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arXiv:2606.12557

New bounds on private simultaneous quantum message passing

Uma Girish, Alex May, Natalie Parham, Henry Yuen

quant-phSubmitted Jun 10, 2026• Announced Jun 12, 2026• 0 citations• Revised Jul 3, 2026 (v2)
v1 · Jun 10, 2026v2 · Jul 3, 2026
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Abstract

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.

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Concrete Functions in Private Simultaneous Quantum Messages: Quantum Advantages and Structural Limitations

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Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on $n$-bit inputs, which determines whether the players' $n$-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication $(1.633\ldots)n$ qubits, whereas every perfectly correct protocol without prior entanglement requires at least $3n$ qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication $3\log_2 3$. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every $k\geq 3$, we construct a perfectly correct and perfectly private one-shot protocol for $k$-bit AND using one qudit of dimension at most $k$ per player, and hence at most $k\log_2 k$ qubits in total.

quant-pharXiv:2610.23090•Oct 19, 2026•0 citations

Ideal Private Simultaneous Messages Schemes and Their Applications

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Private Simultaneous Messages (PSM) is a minimal model for secure computation, where two parties, Alice and Bob, have private inputs x,y and a shared random string. Each of them sends a single message to an external party, Charlie, who can compute f(x,y) for a public function f but learns nothing else. The problem of narrowing the gap between upper and lower bounds on the communication complexity of PSM has been widely studied, but the gap still remains exponential. In this work, we study the communication complexity of PSM from a different perspective and introduce a special class of PSM, referred to as ideal PSM, in which each party’s message length attains the minimum, that is, their messages are taken from the same domain as inputs. We initiate a systematic study of ideal PSM with a complete characterization, several positive results, and applications. First, we provide a characterization of the class of functions that admit ideal PSM, based on permutation groups acting on the input domain. This characterization allows us to derive asymptotic upper bounds on the total number of such functions and a complete list for small domains. We also present several infinite families of functions of practical interest that admit ideal PSM. Interestingly, by simply restricting the input domains of these ideal PSM schemes, we can recover most of the existing PSM schemes that achieve the best known communication complexity in various computation models. As applications, we show that these ideal PSM schemes yield novel communication-efficient PSM schemes for functions with sparse or dense truth-tables and those with low-rank truth-tables. Furthermore, we obtain a PSM scheme for general functions that improves the constant factor in the dominant term of the best known communication complexity. An additional advantage is that our scheme simplifies the existing construction by avoiding the hierarchical design of internally invoking PSM schemes for smaller functions.

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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 $c>0$.

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We study the complexity of quantum decomposable randomized encodings (QDRE). We establish a bi-directional connection between the QDRE model and the communication complexity model of quantum private simultaneous message protocols (QPSM), where classical communication is free, and quantum communication is the primary complexity measure. We show the following upper and lower bounds. (1) For every quantum channel mapping $n$ to $m$ qubits, we give a QPSM protocol with quantum communication $m$, using exponential pre-shared entanglement. In particular, constant-output channels have $O(1)$ quantum communication, and classical-output channels require no quantum communication. (2) With $O(n)$ pre-shared entanglement, every one-qubit-output channel has an $O(n)$-quantum-communication QPSM protocol. Conversely, there exists a channel and a constant $c>0$ for which any protocol with at most $cn$ pre-shared entanglement requires $Ω(n)$ quantum communication. (3) In contrast, we identify a structured class of quantum channels, Clifford-induced channels, for which there exist QPSM protocols with $O(n)$ pre-shared entangled qubits and only $O(1)$ quantum communication complexity. Through our connection, we obtain analogous upper and lower bounds on the quantum encoding size of QDRE. Our results show that the amount of pre-shared entanglement plays a central role in quantum encoding size and quantum communication complexity.

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Private simultaneous messages based on quadratic residues

Kazumasa Shinagawa, Reo Eriguchi, Shohei Satake +1 more

Abstract Private Simultaneous Messages (PSM) model is a minimal model for secure multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai (Cryptology and Information Security Series 2013) constructed PSM protocols based on quadratic residues. In this paper, we define QR-PSM protocols as a generalization of these protocols. A QR-PSM protocol is a PSM protocol whose decoding function outputs the quadratic residuosity modulo p of what is computed from messages. We design a QR-PSM protocol for any symmetric function $$f: \{0,1\}^n \rightarrow \{0,1\}$$ f : { 0 , 1 } n → { 0 , 1 } of communication complexity $$O(n^2)$$ O ( n 2 ) . As far as we know, it is the most efficient PSM protocol for symmetric functions since the previously known best PSM protocol was of $$O(n^2\log n)$$ O ( n 2 log n ) (Beimel et al., CRYPTO 2014). We also study the sizes of the underlying finite fields $$\mathbb {F}_p$$ F p in the protocols since the communication complexity of a QR-PSM protocol is proportional to the bit length of the prime p. We show that there is a prime $$p \le (1+o(1))N^22^{2N-2}$$ p ≤ ( 1 + o ( 1 ) ) N 2 2 2 N - 2 such that any length-N pattern of quadratic (non)residues appears modulo p (and hence it can be used for general QR-PSM protocols), which improves the Peralta’s known result (Mathematics of Computation 1992) by a constant factor $$(1+\sqrt{2})^2$$ ( 1 + 2 ) 2 .

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Exponential lower bounds for low-degree strategies in position-based quantum cryptography

Isabel M. Moreno-Cuadrado, David Perez-Garcia, Carlos Palazuelos

Non-local quantum computation (NLQC) consists on the implementation of a bipartite unitary $U_{AB}$ by two cooperating distant players by means of a two-round protocol with a single intermediate round of simultaneous communication. It is a major open question to know whether there exists a unitary $U_{AB}$ whose implementation in NLQC requires extra quantum systems with dimensions ${\rm exp}(n^{Ω(1)})$, where $n$ is the dimension of systems $A$ and $B$. This type of exponential lower bound is essential for security of position-based quantum cryptography, since the capabilities of the adversaries are precisely described by NLQC. It has also been connected to many other problems, such as the optimality of universal quantum simulators, or the expected properties of holographic quantum gravity. Currently the best lower bounds are only sublinear in $n$. In this paper we consider the family of diagonal ${\pm 1}$ valued unitaries $U_{\varepsilon}$, indexed by an element of the Boolean hypercube $\varepsilon \in \{\pm 1\}^{n^2}$, i.e. $U_{\varepsilon}|i\rangle_A|j\rangle_B =\varepsilon_{ij} |i\rangle_A|j\rangle_B$. We show that if $U_{\varepsilon}$ can be implemented in NLQC with constant accuracy for all $\varepsilon$, and the dependency on $\varepsilon$ in the first round of the associated strategy is a polynomial of degree $O(n^{\frac{1}{4}-δ})$, then the implementation of at least one $U_{\varepsilon}$ in NLQC requires resources scaling as $\exp\left(Ω\left(\frac{n^{2δ}}{(\log n)^2}\right)\right)$. The proof is based on geometric properties of particular Banach spaces, namely their type constants, together with random estimates of Boolean functions valued on them.

quant-pharXiv:2609.35592•Sep 29, 2026•0 citations
In the private simultaneous message (PSM) setting, kkk players obtain inputs xi∈{0,1}nx_i\in\{0,1\}^nxi​∈{0,1}n and then each send messages to a referee, who should learn f(x1,...,xk)f(x_1,...,x_k)f(x1​,...,xk​) but no other information about (x1,...,xk)(x_1,...,x_k)(x1​,...,xk​). 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 kkk-player quantum PSM with perfect correctness. This leads to quadratic lower bounds for explicit functions. 2) The rank of the communication matrix of f(x1,x2)f(x_1,x_2)f(x1​,x2​) 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 sss be the size of a quantum circuit computing fff, dfd_fdf​ be the circuit depth, kkk the number of players, nnn the number of bits received by each player, and εεε a correctness parameter, we obtain PSMk∗(f)≤(kn+s)⋅log⁡O(df)(s/ε)\mathsf{PSM}_k^*(f) \leq (kn +s) \cdot \log^{O(d_f)}(s/ε)PSMk∗​(f)≤(kn+s)⋅logO(df​)(s/ε). 2) The square of the Fourier 1 norm of fff, ∥f^∥12\Vert \hat{f}\Vert_1^2∥f^​∥12​, upper bounds the classical PSM complexity, PSM(f)≤O(∥f^∥12)\mathsf{PSM}(f)\leq O(\Vert \hat{f} \Vert^2_1)PSM(f)≤O(∥f^​∥12​). In proving the first upper bound, we generalize existing TTT-depth based techniques for NLQC from 222 to k≥2k\geq 2k≥2 parties, and consider cases where the Clifford layers are restricted to having small light cones.
Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on nnn-bit inputs, which determines whether the players' nnn-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication (1.633…)n(1.633\ldots)n(1.633…)n qubits, whereas every perfectly correct protocol without prior entanglement requires at least 3n3n3n qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication 3log⁡233\log_2 33log2​3. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every k≥3k\geq 3k≥3, we construct a perfectly correct and perfectly private one-shot protocol for kkk-bit AND using one qudit of dimension at most kkk per player, and hence at most klog⁡2kk\log_2 kklog2​k qubits in total.
Private Simultaneous Messages (PSM) is a minimal model for secure computation, where two parties, Alice and Bob, have private inputs x,y and a shared random string. Each of them sends a single message to an external party, Charlie, who can compute f(x,y) for a public function f but learns nothing else. The problem of narrowing the gap between upper and lower bounds on the communication complexity of PSM has been widely studied, but the gap still remains exponential. In this work, we study the communication complexity of PSM from a different perspective and introduce a special class of PSM, referred to as ideal PSM, in which each party’s message length attains the minimum, that is, their messages are taken from the same domain as inputs. We initiate a systematic study of ideal PSM with a complete characterization, several positive results, and applications. First, we provide a characterization of the class of functions that admit ideal PSM, based on permutation groups acting on the input domain. This characterization allows us to derive asymptotic upper bounds on the total number of such functions and a complete list for small domains. We also present several infinite families of functions of practical interest that admit ideal PSM. Interestingly, by simply restricting the input domains of these ideal PSM schemes, we can recover most of the existing PSM schemes that achieve the best known communication complexity in various computation models. As applications, we show that these ideal PSM schemes yield novel communication-efficient PSM schemes for functions with sparse or dense truth-tables and those with low-rank truth-tables. Furthermore, we obtain a PSM scheme for general functions that improves the constant factor in the dominant term of the best known communication complexity. An additional advantage is that our scheme simplifies the existing construction by avoiding the hierarchical design of internally invoking PSM schemes for smaller functions.
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 kkk players receives an unrestricted string xj∈{0,1}nx_j\in\{0,1\}^nxj​∈{0,1}n, and the referee must decide whether x1⊕⋯⊕xk=0nx_1\oplus\cdots\oplus x_k=0^nx1​⊕⋯⊕xk​=0n. For k≥8k\ge8k≥8 and worst-case error at most 1/31/31/3, we prove quantum lower bounds of Ω(min⁡{n,k2})Ω(\min\{n,k^2\})Ω(min{n,k2}) on total communication and Ω(min⁡{n/k,k})Ω(\min\{n/k,k\})Ω(min{n/k,k}) on maximum message length. A classical public-coin protocol uses two bits per player with error at most 1/41/41/4. For n=k2n=k^2n=k2 and worst-case error at most 1/31/31/3, we prove that quantum protocols require maximum message length Ω(n)Ω(\sqrt n)Ω(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 kkk-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(n1−1/k)Ω(n^{1-1/k})Ω(n1−1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error ε∈(0,1/8)\varepsilon\in(0,1/8)ε∈(0,1/8), we also prove a bounded-error quantum lower bound of Ω(n(k−1)/(k+1))Ω(n^{(k-1)/(k+1)})Ω(n(k−1)/(k+1)) while we show that public-coin protocols solve the relation unambiguously with maximum message length O(log⁡n)O(\log n)O(logn), giving exponential separations in both regimes. The implicit constants are independent of kkk, and both quantum lower bounds become linear when k≥clog⁡nk\ge c\log nk≥clogn for any fixed c>0c>0c>0.
We study the complexity of quantum decomposable randomized encodings (QDRE). We establish a bi-directional connection between the QDRE model and the communication complexity model of quantum private simultaneous message protocols (QPSM), where classical communication is free, and quantum communication is the primary complexity measure. We show the following upper and lower bounds. (1) For every quantum channel mapping nnn to mmm qubits, we give a QPSM protocol with quantum communication mmm, using exponential pre-shared entanglement. In particular, constant-output channels have O(1)O(1)O(1) quantum communication, and classical-output channels require no quantum communication. (2) With O(n)O(n)O(n) pre-shared entanglement, every one-qubit-output channel has an O(n)O(n)O(n)-quantum-communication QPSM protocol. Conversely, there exists a channel and a constant c>0c>0c>0 for which any protocol with at most cncncn pre-shared entanglement requires Ω(n)Ω(n)Ω(n) quantum communication. (3) In contrast, we identify a structured class of quantum channels, Clifford-induced channels, for which there exist QPSM protocols with O(n)O(n)O(n) pre-shared entangled qubits and only O(1)O(1)O(1) quantum communication complexity. Through our connection, we obtain analogous upper and lower bounds on the quantum encoding size of QDRE. Our results show that the amount of pre-shared entanglement plays a central role in quantum encoding size and quantum communication complexity.
Abstract Private Simultaneous Messages (PSM) model is a minimal model for secure multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai (Cryptology and Information Security Series 2013) constructed PSM protocols based on quadratic residues. In this paper, we define QR-PSM protocols as a generalization of these protocols. A QR-PSM protocol is a PSM protocol whose decoding function outputs the quadratic residuosity modulo p of what is computed from messages. We design a QR-PSM protocol for any symmetric function f:{0,1}n→{0,1}f: \{0,1\}^n \rightarrow \{0,1\}f:{0,1}n→{0,1} f : { 0 , 1 } n → { 0 , 1 } of communication complexity O(n2)O(n^2)O(n2) O ( n 2 ) . As far as we know, it is the most efficient PSM protocol for symmetric functions since the previously known best PSM protocol was of O(n2log⁡n)O(n^2\log n)O(n2logn) O ( n 2 log n ) (Beimel et al., CRYPTO 2014). We also study the sizes of the underlying finite fields Fp\mathbb {F}_pFp​ F p in the protocols since the communication complexity of a QR-PSM protocol is proportional to the bit length of the prime p. We show that there is a prime p≤(1+o(1))N222N−2p \le (1+o(1))N^22^{2N-2}p≤(1+o(1))N222N−2 p ≤ ( 1 + o ( 1 ) ) N 2 2 2 N - 2 such that any length-N pattern of quadratic (non)residues appears modulo p (and hence it can be used for general QR-PSM protocols), which improves the Peralta’s known result (Mathematics of Computation 1992) by a constant factor (1+2)2(1+\sqrt{2})^2(1+2​)2 ( 1 + 2 ) 2 .
Non-local quantum computation (NLQC) consists on the implementation of a bipartite unitary UABU_{AB}UAB​ by two cooperating distant players by means of a two-round protocol with a single intermediate round of simultaneous communication. It is a major open question to know whether there exists a unitary UABU_{AB}UAB​ whose implementation in NLQC requires extra quantum systems with dimensions exp(nΩ(1)){\rm exp}(n^{Ω(1)})exp(nΩ(1)), where nnn is the dimension of systems AAA and BBB. This type of exponential lower bound is essential for security of position-based quantum cryptography, since the capabilities of the adversaries are precisely described by NLQC. It has also been connected to many other problems, such as the optimality of universal quantum simulators, or the expected properties of holographic quantum gravity. Currently the best lower bounds are only sublinear in nnn. In this paper we consider the family of diagonal ±1{\pm 1}±1 valued unitaries UεU_{\varepsilon}Uε​, indexed by an element of the Boolean hypercube ε∈{±1}n2\varepsilon \in \{\pm 1\}^{n^2}ε∈{±1}n2, i.e. Uε∣i⟩A∣j⟩B=εij∣i⟩A∣j⟩BU_{\varepsilon}|i\rangle_A|j\rangle_B =\varepsilon_{ij} |i\rangle_A|j\rangle_BUε​∣i⟩A​∣j⟩B​=εij​∣i⟩A​∣j⟩B​. We show that if UεU_{\varepsilon}Uε​ can be implemented in NLQC with constant accuracy for all ε\varepsilonε, and the dependency on ε\varepsilonε in the first round of the associated strategy is a polynomial of degree O(n14−δ)O(n^{\frac{1}{4}-δ})O(n41​−δ), then the implementation of at least one UεU_{\varepsilon}Uε​ in NLQC requires resources scaling as exp⁡(Ω(n2δ(log⁡n)2))\exp\left(Ω\left(\frac{n^{2δ}}{(\log n)^2}\right)\right)exp(Ω((logn)2n2δ​)). The proof is based on geometric properties of particular Banach spaces, namely their type constants, together with random estimates of Boolean functions valued on them.