Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.26425

Pseudodeterminism and MA != NP^BPP in Communication Complexity

Thomas W. Watson

Published Aug 26, 2026• 0 citations
arXivHTMLPDF

Abstract

We prove an exponential separation between zero-sided-error randomized and two-sided-error pseudodeterministic communication complexities of partial boolean functions. This qualitatively improves and simplifies the proof of the separation by Göös, Harms, Riazanov, Sofronova, Sokolov, and Yuan (STOC 2026), which had a two-sided-error randomized upper bound. Then, we generalize our technique to separate the communication complexity analogues of MA and NP^BPP.

DOI
10.48550/arxiv.2608.26425

Related papers

Pseudodeterministic Communication Complexity

Mika Göös, Nathaniel Harms, Artur Riazanov +3 more

We exhibit an n-bit partial function with randomized communication complexity O(logn) but such that any completion of this function into a total one requires randomized communication complexity nΩ(1). In particular, this shows an exponential separation between randomized and pseudodeterministic communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.

ACM Symposium on Theory of Computing (STOC)•Jun 9, 2026•0 citations

Pseudodeterministic Communication Complexity

Mika Göös, Harms, Nathaniel, Riazanov, Artur +3 more

We exhibit an $n$-bit partial function with randomized communication complexity $O(\log n)$ but such that any completion of this function into a total one requires randomized communication complexity $n^{Ω(1)}$. In particular, this shows an exponential separation between randomized and \emph{pseudodeterministic} communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.

arXiv:2511.04794•Nov 6, 2025•0 citations

One-Way Communication Complexity of Partial XOR Functions

Vladimir V. Podolskii, Sluch, Dmitrii

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

arXiv:2310.20606•Oct 31, 2023•0 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.

arXiv:1911.02218•Computational Complexity•Oct 28, 2022•8 citations

Quantum-Classical Equivalence for AND-Functions

Sreejata Kishor Bhattacharya, Farzan Byramji, Arkadev Chattopadhyay +2 more

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), $$ when the outer function $f$ is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function $f$, the bounded-error quantum and classical deterministic communication complexities of the function $f \circ \mathrm{AND}_2$ are polynomially related, up to polylogarithmic factors in $n$. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of $f$. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.

cs.CCCross-listedarXiv:2606.03249•Jun 3, 2026•0 citations

Near-Optimal Separations of Certificate Complexity from Randomized and Quantum Query Complexity

Andris Ambainis, Jānis Iraids, Martins Kokainis

We study how large the certificate complexity ${C}(f)$ of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function $f$ whose randomized query complexity with one-sided error satisfies ${R}_1(f) = Θ(\sqrt{{C}(f)})$. This separation is optimal even when two-sided error is allowed. From this construction, we obtain another total Boolean function $F$ with bounded-error quantum query complexity ${Q}(F) = \widetilde O({C}(F)^{1/4})$, attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are $\widetilde O(\sqrt{{C}(F)})$. We also prove matching lower bounds for these two measures on $F$.

quant-pharXiv:2609.11664•Sep 11, 2026•0 citations
We prove an exponential separation between zero-sided-error randomized and two-sided-error pseudodeterministic communication complexities of partial boolean functions. This qualitatively improves and simplifies the proof of the separation by Göös, Harms, Riazanov, Sofronova, Sokolov, and Yuan (STOC 2026), which had a two-sided-error randomized upper bound. Then, we generalize our technique to separate the communication complexity analogues of MA and NP^BPP.
We exhibit an n-bit partial function with randomized communication complexity O(logn) but such that any completion of this function into a total one requires randomized communication complexity nΩ(1). In particular, this shows an exponential separation between randomized and pseudodeterministic communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.
We exhibit an nnn-bit partial function with randomized communication complexity O(log⁡n)O(\log n)O(logn) but such that any completion of this function into a total one requires randomized communication complexity nΩ(1)n^{Ω(1)}nΩ(1). In particular, this shows an exponential separation between randomized and \emph{pseudodeterministic} communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.
Boolean function F(x,y)F(x,y)F(x,y) for x,y∈{0,1}nx,y \in \{0,1\}^nx,y∈{0,1}n is an XOR function if F(x,y)=f(x⊕y)F(x,y)=f(x\oplus y)F(x,y)=f(x⊕y) for some function fff on nnn input bits, where ⊕\oplus⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of FFF is closely related to parity decision tree complexity of fff. Montanaro and Osbourne (2009) observed that one-sided communication complexity Dcc→(F)D_{cc}^{\rightarrow}(F)Dcc→​(F) of FFF is exactly equal to nonadaptive parity decision tree complexity NADT⊕(f)NADT^{\oplus}(f)NADT⊕(f) of fff. Hatami et al. (2018) showed that unrestricted communication complexity of FFF is polynomially related to parity decision tree complexity of fff. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of fff. On the one hand, if Dcc→(F)=tD_{cc}^{\rightarrow}(F)=tDcc→​(F)=t and fff is undefined on at most O(2n−tn−t)O(\frac{2^{n-t}}{\sqrt{n-t}})O(n−t​2n−t​), then NADT⊕(f)=tNADT^{\oplus}(f)=tNADT⊕(f)=t. On the other hand, for a wide range of values of Dcc→(F)D_{cc}^{\rightarrow}(F)Dcc→​(F) and NADT⊕(f)NADT^{\oplus}(f)NADT⊕(f) (from constant to n−2n-2n−2) we provide partial functions for which Dcc→(F)<NADT⊕(f)D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)Dcc→​(F)<NADT⊕(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.
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 Ω~(N1/4)\tilde{\Omega}\left(N^{1/4}\right)Ω~(N1/4) . 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 Ω(N1/2)\Omega\left(N^{1/2}\right)Ω(N1/2) 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.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form F(x,y)=f(x1∧y1,…,xn∧yn), F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), F(x,y)=f(x1​∧y1​,…,xn​∧yn​), when the outer function fff is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function fff, the bounded-error quantum and classical deterministic communication complexities of the function f∘AND2f \circ \mathrm{AND}_2f∘AND2​ are polynomially related, up to polylogarithmic factors in nnn. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of fff. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.
We study how large the certificate complexity C(f){C}(f)C(f) of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function fff whose randomized query complexity with one-sided error satisfies R1(f)=Θ(C(f)){R}_1(f) = Θ(\sqrt{{C}(f)})R1​(f)=Θ(C(f)​). This separation is optimal even when two-sided error is allowed. From this construction, we obtain another total Boolean function FFF with bounded-error quantum query complexity Q(F)=O~(C(F)1/4){Q}(F) = \widetilde O({C}(F)^{1/4})Q(F)=O(C(F)1/4), attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are O~(C(F))\widetilde O(\sqrt{{C}(F)})O(C(F)​). We also prove matching lower bounds for these two measures on FFF.