arXiv:2502.13427

Does there exist a quantum fingerprinting protocol without coherent measurements?

Atsuya Hasegawa, Srijita Kundu, François Le Gall, Harumichi Nishimura, Qisheng Wang

Published Feb 19, 2025• 0 citations

Abstract

Buhrman, Cleve, Watrous, and de Wolf (PRL 2001) discovered the quantum fingerprinting protocol, which is the quantum SMP protocol with $O(\log n)$ qubits communication for the equality problem. In the protocol, Alice and Bob create some quantum fingerprints of their inputs, and the referee conducts the SWAP tests for the quantum fingerprints. Since $Ω(\sqrt{n})$ bits communication is required with the classical SMP scheme for the equality problem first shown by Newman and Szegedy (STOC 1996), there exists an exponential quantum advantage in the amount of communication. In this paper, we consider a setting in which the referee can do only incoherent measurements rather than coherent measurements including the SWAP tests. We first show that, in the case of one-way LOCC measurements, $Ω(\sqrt{n})$ qubits communication is required. To prove the result, we derive a new method to replace quantum messages by classical messages and consider a reduction to the optimal lower bound in the hybrid SMP model where one message is quantum and the other is classical, which was first shown by Klauck and Podder (MFCS 2014). Our method uses the result of Oszmaniec, Guerini, Wittek, and Acín (PRL 2017), who showed that general POVM measurements can be simulated by randomized projective measurements with small ancilla qubits, and Newman's theorem. We further investigate the setting of quantum SMP protocols with two-way LOCC measurements, and derive a lower bound against some restricted two-way LOCC measurements. To prove it, we revisit the technique to replace quantum messages by classical deterministic messages introduced by Aaronson (ToC 2005) and generalized by Gavinsky, Regev, and de Wolf (CJTCS 2008), and show that, using the deterministic message, the referee can simulate the two-way LOCC measurements.

DOI: 10.48550/arxiv.2502.13427

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Daiki Suruga

quant-pharXiv:2609.14658Sep 15, 20260 citations

On the Limits of Quantum Multiparty Simultaneous Communication

Pedro Montealegre, Ivan Rapaport, Jorge Valenzuela

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New bounds on private simultaneous quantum message passing

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Quantum versus Randomized Communication Complexity, with Efficient Players

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Multi-party quantum fingerprinting with weak coherent pulses: circuit design and protocol analysis

Hipólito Gómez-Sousa

arXiv:2006.00708New Journal of PhysicsOct 20, 20201 citation

Experimental demonstration of quantum advantage in communication complexity for Euclidean distance problem

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arXiv:2502.13427

Does there exist a quantum fingerprinting protocol without coherent measurements?

Atsuya Hasegawa, Srijita Kundu, François Le Gall, Harumichi Nishimura, Qisheng Wang

Published Feb 19, 2025• 0 citations

arXiv HTML PDF

Abstract

Buhrman, Cleve, Watrous, and de Wolf (PRL 2001) discovered the quantum fingerprinting protocol, which is the quantum SMP protocol with

O(\log n)

qubits communication for the equality problem. In the protocol, Alice and Bob create some quantum fingerprints of their inputs, and the referee conducts the SWAP tests for the quantum fingerprints. Since

Ω(\sqrt{n})

bits communication is required with the classical SMP scheme for the equality problem first shown by Newman and Szegedy (STOC 1996), there exists an exponential quantum advantage in the amount of communication. In this paper, we consider a setting in which the referee can do only incoherent measurements rather than coherent measurements including the SWAP tests. We first show that, in the case of one-way LOCC measurements,

Ω(\sqrt{n})

qubits communication is required. To prove the result, we derive a new method to replace quantum messages by classical messages and consider a reduction to the optimal lower bound in the hybrid SMP model where one message is quantum and the other is classical, which was first shown by Klauck and Podder (MFCS 2014). Our method uses the result of Oszmaniec, Guerini, Wittek, and Acín (PRL 2017), who showed that general POVM measurements can be simulated by randomized projective measurements with small ancilla qubits, and Newman's theorem. We further investigate the setting of quantum SMP protocols with two-way LOCC measurements, and derive a lower bound against some restricted two-way LOCC measurements. To prove it, we revisit the technique to replace quantum messages by classical deterministic messages introduced by Aaronson (ToC 2005) and generalized by Gavinsky, Regev, and de Wolf (CJTCS 2008), and show that, using the deterministic message, the referee can simulate the two-way LOCC measurements.

DOI: 10.48550/arxiv.2502.13427

Related papers

Exact Asymptotic Rates and an Exponential Strong Converse for quantum SMP and One-Way Communication

Daiki Suruga

Computing many instances jointly can reduce communication per instance. We ask whether such savings occur in the simultaneous-message-passing (SMP) model and how they depend on quantum messages and shared resources. For every finite total function $f$ and fixed error $0\le\varepsilon<1$ , we prove that the optimal worst-case communication per instance for $f^n$ converges to $D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|,$ where $\mathrm{Row}(f)$ and $\mathrm{Col}(f)$ are the sets of distinct rows and columns of the table $(f(x,y))_{x,y}$ . This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all $n$ outputs correctly to decay as $2^{-Ω(n)}$ , even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is $(1-\varepsilon)D^\parallel(f)$ . Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with $D^\parallel(f)$ replaced by $\log|\mathrm{Row}(f)|$ . Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.Computing many instances jointly can reduce communication per instance. We ask whether such savings occur in the simultaneous-message-passing (SMP) model and how they depend on quantum messages and shared resources. For every finite total function $f$ and fixed error $0\le\varepsilon<1$, we prove that the optimal worst-case communication per instance for $f^n$ converges to \[ D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|, \] where $\mathrm{Row}(f)$ and $\mathrm{Col}(f)$ are the sets of distinct rows and columns of the table $(f(x,y))_{x,y}$. This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all $n$ outputs correctly to decay as $2^{-Ω(n)}$, even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is $(1-\varepsilon)D^\parallel(f)$. Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with $D^\parallel(f)$ replaced by $\log|\mathrm{Row}(f)|$. Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.

quant-pharXiv:2609.14658Sep 15, 20260 citations

On the Limits of Quantum Multiparty Simultaneous Communication

Pedro Montealegre, Ivan Rapaport, Jorge Valenzuela

We study the relative power of public randomness and quantum communication in the multiparty Simultaneous Message Passing (SMP) model. By analyzing a total Boolean function and a relational problem, we establish polynomial and exponential separations between classical public-coin protocols and quantum protocols without shared randomness or prior entanglement. We first introduce XOR Equality, a total Boolean function in which each of $k$ players receives an unrestricted string $x_j\in\{0,1\}^n$ , and the referee must decide whether $x_1\oplus\cdots\oplus x_k=0^n$ . For $k\ge8$ and worst-case error at most $1/3$ , we prove quantum lower bounds of $Ω(\min\{n,k^2\})$ on total communication and $Ω(\min\{n/k,k\})$ on maximum message length. A classical public-coin protocol uses two bits per player with error at most $1/4$ . For $n=k^2$ and worst-case error at most $1/3$ , we prove that quantum protocols require maximum message length $Ω(\sqrt n)$ qubits, whereas two public-coin bits per player suffice. We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a $k$ -player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of $Ω(n^{1-1/k})$ on maximum message length, matched by a classical private-coin protocol. For any fixed error $\varepsilon\in(0,1/8)$ , we also prove a bounded-error quantum lower bound of $Ω(n^{(k-1)/(k+1)})$ while we show that public-coin protocols solve the relation unambiguously with maximum message length $O(\log n)$ , giving exponential separations in both regimes. The implicit constants are independent of $k$ , and both quantum lower bounds become linear when $k\ge c\log n$ for any fixed $c>0$ .We study the relative power of public randomness and quantum communication in the multiparty Simultaneous Message Passing (SMP) model. By analyzing a total Boolean function and a relational problem, we establish polynomial and exponential separations between classical public-coin protocols and quantum protocols without shared randomness or prior entanglement. We first introduce XOR Equality, a total Boolean function in which each of $k$ players receives an unrestricted string $x_j\in\{0,1\}^n$, and the referee must decide whether $x_1\oplus\cdots\oplus x_k=0^n$. For $k\ge8$ and worst-case error at most $1/3$, we prove quantum lower bounds of $Ω(\min\{n,k^2\})$ on total communication and $Ω(\min\{n/k,k\})$ on maximum message length. A classical public-coin protocol uses two bits per player with error at most $1/4$. For $n=k^2$ and worst-case error at most $1/3$, we prove that quantum protocols require maximum message length $Ω(\sqrt n)$ qubits, whereas two public-coin bits per player suffice. We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a $k$-player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of $Ω(n^{1-1/k})$ on maximum message length, matched by a classical private-coin protocol. For any fixed error $\varepsilon\in(0,1/8)$, we also prove a bounded-error quantum lower bound of $Ω(n^{(k-1)/(k+1)})$ while we show that public-coin protocols solve the relation unambiguously with maximum message length $O(\log n)$, giving exponential separations in both regimes. The implicit constants are independent of $k$, and both quantum lower bounds become linear when $k\ge c\log n$ for any fixed $c>0$.

cs.CCCross-listedarXiv:2609.10289Sep 10, 20260 citations

New bounds on private simultaneous quantum message passing

Uma Girish, Alex May, Natalie Parham +1 more

In the private simultaneous message (PSM) setting, $k$ players obtain inputs $x_i\in\{0,1\}^n$ and then each send messages to a referee, who should learn $f(x_1,...,x_k)$ but no other information about $(x_1,...,x_k)$ . The PSM setting was introduced as a minimal model for secure multiparty computation and has connections to Boolean function complexity. In the quantum setting, PSM has been related to non-local quantum computation (NLQC). The communication and correlation cost of implementing PSM remains poorly understood. Here, we give new upper and lower bounds on the (quantum) PSM model. For lower bounds, we show: 1) Nečiporuk's measure lower bounds the entanglement required for $k$ -player quantum PSM with perfect correctness. This leads to quadratic lower bounds for explicit functions. 2) The rank of the communication matrix of $f(x_1,x_2)$ lower bounds 2-player quantum PSM with perfect privacy but imperfect correctness. This implies a previously unknown lower bound on classical PSM with imperfect correctness. When allowing quantum communication and shared entanglement, these are the first lower bounds on quantum PSM that make use of the privacy condition. For upper bounds, we show: 1) Letting $s$ be the size of a quantum circuit computing $f$ , $d_f$ be the circuit depth, $k$ the number of players, $n$ the number of bits received by each player, and $ε$ a correctness parameter, we obtain $\mathsf{PSM}_k^*(f) \leq (kn +s) \cdot \log^{O(d_f)}(s/ε)$ . 2) The square of the Fourier 1 norm of $f$ , $\Vert \hat{f}\Vert_1^2$ , upper bounds the classical PSM complexity, $\mathsf{PSM}(f)\leq O(\Vert \hat{f} \Vert^2_1)$ . In proving the first upper bound, we generalize existing $T$ -depth based techniques for NLQC from $2$ to $k\geq 2$ parties, and consider cases where the Clifford layers are restricted to having small light cones.In the private simultaneous message (PSM) setting, $k$ players obtain inputs $x_i\in\{0,1\}^n$ and then each send messages to a referee, who should learn $f(x_1,...,x_k)$ but no other information about $(x_1,...,x_k)$. The PSM setting was introduced as a minimal model for secure multiparty computation and has connections to Boolean function complexity. In the quantum setting, PSM has been related to non-local quantum computation (NLQC). The communication and correlation cost of implementing PSM remains poorly understood. Here, we give new upper and lower bounds on the (quantum) PSM model. For lower bounds, we show: 1) Nečiporuk's measure lower bounds the entanglement required for $k$-player quantum PSM with perfect correctness. This leads to quadratic lower bounds for explicit functions. 2) The rank of the communication matrix of $f(x_1,x_2)$ lower bounds 2-player quantum PSM with perfect privacy but imperfect correctness. This implies a previously unknown lower bound on classical PSM with imperfect correctness. When allowing quantum communication and shared entanglement, these are the first lower bounds on quantum PSM that make use of the privacy condition. For upper bounds, we show: 1) Letting $s$ be the size of a quantum circuit computing $f$, $d_f$ be the circuit depth, $k$ the number of players, $n$ the number of bits received by each player, and $ε$ a correctness parameter, we obtain $\mathsf{PSM}_k^*(f) \leq (kn +s) \cdot \log^{O(d_f)}(s/ε)$. 2) The square of the Fourier 1 norm of $f$, $\Vert \hat{f}\Vert_1^2$, upper bounds the classical PSM complexity, $\mathsf{PSM}(f)\leq O(\Vert \hat{f} \Vert^2_1)$. In proving the first upper bound, we generalize existing $T$-depth based techniques for NLQC from $2$ to $k\geq 2$ parties, and consider cases where the Clifford layers are restricted to having small light cones.

quant-pharXiv:2606.12557Jun 12, 20260 citations

Quantum versus Randomized Communication Complexity, with Efficient Players

Uma Girish, Ran Raz, Avishay Tal

We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient , in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity Gavinsky (2016) and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N , such that: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least $\tilde{\Omega}\left(N^{1/4}\right)$ . All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of $\Omega\left(N^{1/2}\right)$ in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient , in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity Gavinsky (2016) and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N , such that: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least $\tilde{\Omega}\left(N^{1/4}\right)$ . All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of $\Omega\left(N^{1/2}\right)$ in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.

arXiv:1911.02218Computational ComplexityOct 28, 20228 citations

Multi-party quantum fingerprinting with weak coherent pulses: circuit design and protocol analysis

Hipólito Gómez-Sousa

Abstract Quantum communication has been leading the way of many remarkable theoretical results and experimental tests in physics. In this context, quantum communication complexity (QCC) has recently drawn earnest research attention as a tool to optimize the amounts of transmitted qubits and energy that are required to implement distributed computational tasks. On this matter, we introduce a novel multi-user quantum fingerprinting (QF) protocol that is ready to be implemented with existing technology. Particularly, we extend to the multi-user framework a well-known two-user coherent-state fingerprinting scheme. This generalization is highly non-trivial for a twofold reason, as it requires not only to extend the set of protocol rules but also to specify a procedure for designing the optical devices intended for the generalized protocol. Much of the importance of our work arises from the fact that the obtained QCC figures of merit allow direct comparison with the best-known classical multi-user fingerprinting protocol, of significance in the field of computer technologies and networking. Furthermore, as one of the main contributions of the manuscript, we deduce innovative analytical upper bounds on the amount of transmitted quantum information that are even valid in the two-user protocol as a particular case. These original analytical bounds are of interest for estimating the realistic protocol performance prior to experimental realizations. Ultimately, comparative results are provided to contrast different protocol implementation strategies and, importantly, to show that, under realistic circumstances, the multi-user protocol can achieve tasks that are impossible by using classical communication alone. Our work provides relevant contributions towards understanding the nature and the limitations of QF and, on a broader scope, also the limitations and possibilities of quantum-communication networks embracing a node that is accessed by multiple users at the same time.Abstract Quantum communication has been leading the way of many remarkable theoretical results and experimental tests in physics. In this context, quantum communication complexity (QCC) has recently drawn earnest research attention as a tool to optimize the amounts of transmitted qubits and energy that are required to implement distributed computational tasks. On this matter, we introduce a novel multi-user quantum fingerprinting (QF) protocol that is ready to be implemented with existing technology. Particularly, we extend to the multi-user framework a well-known two-user coherent-state fingerprinting scheme. This generalization is highly non-trivial for a twofold reason, as it requires not only to extend the set of protocol rules but also to specify a procedure for designing the optical devices intended for the generalized protocol. Much of the importance of our work arises from the fact that the obtained QCC figures of merit allow direct comparison with the best-known classical multi-user fingerprinting protocol, of significance in the field of computer technologies and networking. Furthermore, as one of the main contributions of the manuscript, we deduce innovative analytical upper bounds on the amount of transmitted quantum information that are even valid in the two-user protocol as a particular case. These original analytical bounds are of interest for estimating the realistic protocol performance prior to experimental realizations. Ultimately, comparative results are provided to contrast different protocol implementation strategies and, importantly, to show that, under realistic circumstances, the multi-user protocol can achieve tasks that are impossible by using classical communication alone. Our work provides relevant contributions towards understanding the nature and the limitations of QF and, on a broader scope, also the limitations and possibilities of quantum-communication networks embracing a node that is accessed by multiple users at the same time.

arXiv:2006.00708New Journal of PhysicsOct 20, 20201 citation

Experimental demonstration of quantum advantage in communication complexity for Euclidean distance problem

Verena Yacoub, Niraj Kumar, Iordanis Kerenidis +1 more

When considering the complexity of communication protocols, the aim is to perform a certain task with the minimum amount of communication resources, such as time and transmitted information. The use of quantum states may lead to an exponential advantage in the use of such resources. Here, we are interested in the task of calculating the Euclidean distance between two vectors representing real data sets. It has been previously shown that it is possible to obtain an advantage for this task based on quantum fingerprinting. This protocol is defined in the simultaneous message passing model of communication complexity, where the two parties do not communicate with each other but send data to a third party, and exploits practical fingerprints generated using trains of coherent state pulses instead of highly entangled qubit states that are hard to generate for large input sizes needed to demonstrate an exponential advantage. We perform a proof-of-principle experimental demonstration of the Euclidean distance protocol using amplitude modulation techniques for encoding non-binary data sets and high-performance superconducting nanowire single-photon detectors required to increase the accessible input size. We show a quantum advantage in transmitted information surpassing the best classical protocol for an input size of $10^8$ , for diverse types of data sets, including those corresponding to real grayscale images, and with reasonable precision and error bounds. Our results highlight the potential of quantum communication complexity for use in a broad set of applications.When considering the complexity of communication protocols, the aim is to perform a certain task with the minimum amount of communication resources, such as time and transmitted information. The use of quantum states may lead to an exponential advantage in the use of such resources. Here, we are interested in the task of calculating the Euclidean distance between two vectors representing real data sets. It has been previously shown that it is possible to obtain an advantage for this task based on quantum fingerprinting. This protocol is defined in the simultaneous message passing model of communication complexity, where the two parties do not communicate with each other but send data to a third party, and exploits practical fingerprints generated using trains of coherent state pulses instead of highly entangled qubit states that are hard to generate for large input sizes needed to demonstrate an exponential advantage. We perform a proof-of-principle experimental demonstration of the Euclidean distance protocol using amplitude modulation techniques for encoding non-binary data sets and high-performance superconducting nanowire single-photon detectors required to increase the accessible input size. We show a quantum advantage in transmitted information surpassing the best classical protocol for an input size of $10^8$, for diverse types of data sets, including those corresponding to real grayscale images, and with reasonable precision and error bounds. Our results highlight the potential of quantum communication complexity for use in a broad set of applications.

quant-pharXiv:2605.31516Jun 1, 20260 citations

Buhrman, Cleve, Watrous, and de Wolf (PRL 2001) discovered the quantum fingerprinting protocol, which is the quantum SMP protocol with

O(\log n)

Ω(\sqrt{n})

Ω(\sqrt{n})

f

and fixed error

0\le\varepsilon<1

, we prove that the optimal worst-case communication per instance for

f^n

converges to

D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|,

where

\mathrm{Row}(f)

and

\mathrm{Col}(f)

are the sets of distinct rows and columns of the table

(f(x,y))_{x,y}

. This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all

n

outputs correctly to decay as

2^{-Ω(n)}

, even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is

(1-\varepsilon)D^\parallel(f)

. Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with

D^\parallel(f)

replaced by

\log|\mathrm{Row}(f)|

. Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.

k

players receives an unrestricted string

x_j\in\{0,1\}^n

, and the referee must decide whether

x_1\oplus\cdots\oplus x_k=0^n

. For

k\ge8

and worst-case error at most

1/3

, we prove quantum lower bounds of

Ω(\min\{n,k^2\})

on total communication and

Ω(\min\{n/k,k\})

on maximum message length. A classical public-coin protocol uses two bits per player with error at most

1/4

. For

n=k^2

and worst-case error at most

1/3

, we prove that quantum protocols require maximum message length

Ω(\sqrt n)

qubits, whereas two public-coin bits per player suffice. We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a

k

-player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of

Ω(n^{1-1/k})

on maximum message length, matched by a classical private-coin protocol. For any fixed error

\varepsilon\in(0,1/8)

, we also prove a bounded-error quantum lower bound of

Ω(n^{(k-1)/(k+1)})

while we show that public-coin protocols solve the relation unambiguously with maximum message length

O(\log n)

, giving exponential separations in both regimes. The implicit constants are independent of

k

, and both quantum lower bounds become linear when

k\ge c\log n

for any fixed

c>0

In the private simultaneous message (PSM) setting,

k

players obtain inputs

x_i\in\{0,1\}^n

and then each send messages to a referee, who should learn

f(x_1,...,x_k)

but no other information about

(x_1,...,x_k)

. The PSM setting was introduced as a minimal model for secure multiparty computation and has connections to Boolean function complexity. In the quantum setting, PSM has been related to non-local quantum computation (NLQC). The communication and correlation cost of implementing PSM remains poorly understood. Here, we give new upper and lower bounds on the (quantum) PSM model. For lower bounds, we show: 1) Nečiporuk's measure lower bounds the entanglement required for

k

-player quantum PSM with perfect correctness. This leads to quadratic lower bounds for explicit functions. 2) The rank of the communication matrix of

f(x_1,x_2)

lower bounds 2-player quantum PSM with perfect privacy but imperfect correctness. This implies a previously unknown lower bound on classical PSM with imperfect correctness. When allowing quantum communication and shared entanglement, these are the first lower bounds on quantum PSM that make use of the privacy condition. For upper bounds, we show: 1) Letting

s

be the size of a quantum circuit computing

f

d_f

be the circuit depth,

k

the number of players,

n

the number of bits received by each player, and

ε

a correctness parameter, we obtain

\mathsf{PSM}_k^*(f) \leq (kn +s) \cdot \log^{O(d_f)}(s/ε)

. 2) The square of the Fourier 1 norm of

f

\Vert \hat{f}\Vert_1^2

, upper bounds the classical PSM complexity,

\mathsf{PSM}(f)\leq O(\Vert \hat{f} \Vert^2_1)

. In proving the first upper bound, we generalize existing

T

-depth based techniques for NLQC from

2

k\geq 2

parties, and consider cases where the Clifford layers are restricted to having small light cones.

\tilde{\Omega}\left(N^{1/4}\right)

. All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of

\Omega\left(N^{1/2}\right)

in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.

10^8

, for diverse types of data sets, including those corresponding to real grayscale images, and with reasonable precision and error bounds. Our results highlight the potential of quantum communication complexity for use in a broad set of applications.