We study how symmetry affects quantum advantage in two-party communication complexity. For any partial function $f$ on length-$n$ strings over a fixed alphabet of size $q$, invariant under simultaneous coordinate permutations, we prove $R^{\mathrm{pub}}(f)=O_q(t\log(2+n/t))$, where $t=\min\{n,Q^{\ast}(f)^2\}$, $R^{\mathrm{pub}}$ denotes public-coin randomized communication complexity, and $Q^{\ast}$ denotes entanglement-assisted quantum communication complexity. This bound is tight for set disjointness and extends the binary-alphabet result of Guan et al. with a smaller logarithmic overhead. The exponent of the logarithmic factor cannot be reduced by any fixed positive amount: for every fixed $\varepsilon\in(0,1)$, there are binary permutation-invariant partial functions with quantum communication complexity $O_\varepsilon(\log\log n)$ and randomized communication complexity $Ω_\varepsilon((\log n)^{1-\varepsilon})$. We also characterize quantum communication complexity by a combinatorial parameter up to a logarithmic factor.
Growing alphabets and graph symmetries admit exponential separations. For every fixed $\varepsilon\in(0,1)$, we exhibit permutation-invariant partial functions on length-$n$ strings over an $n$-symbol alphabet with quantum communication complexity $O_\varepsilon(\log n)$ and randomized communication complexity $Ω_\varepsilon(n^{1-\varepsilon})$. For graph-invariant partial functions on $v$-vertex graphs, we obtain $O_\varepsilon(\log v)$ quantum versus $Ω_\varepsilon(v^{2-\varepsilon})$ randomized communication complexity. All separation protocols require no prior entanglement.
We study how symmetry affects quantum advantage in two-party communication complexity. For any partial function f on length-n strings over a fixed alphabet of size q, invariant under simultaneous coordinate permutations, we prove Rpub(f)=Oq(tlog(2+n/t)), where t=min{n,Q∗(f)2}, Rpub denotes public-coin randomized communication complexity, and Q∗ denotes entanglement-assisted quantum communication complexity. This bound is tight for set disjointness and extends the binary-alphabet result of Guan et al. with a smaller logarithmic overhead. The exponent of the logarithmic factor cannot be reduced by any fixed positive amount: for every fixed ε∈(0,1), there are binary permutation-invariant partial functions with quantum communication complexity Oε(loglogn) and randomized communication complexity Ωε((logn)1−ε). We also characterize quantum communication complexity by a combinatorial parameter up to a logarithmic factor.
Growing alphabets and graph symmetries admit exponential separations. For every fixed ε∈(0,1), we exhibit permutation-invariant partial functions on length-n strings over an n-symbol alphabet with quantum communication complexity Oε(logn) and randomized communication complexity Ωε(n1−ε). For graph-invariant partial functions on v-vertex graphs, we obtain Oε(logv) quantum versus Ωε(v2−ε) randomized communication complexity. All separation protocols require no prior entanglement.
This paper gives a nearly tight characterization of the quantum communication complexity of the permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of the permutation-invariant Boolean functions are quadratically equivalent (up to a logarithmic factor). Our results extend a recent line of research regarding query complexity \cite{AA14, Cha19, BCG+20} to communication complexity, showing symmetry prevents exponential quantum speedups. Furthermore, we show the Log-rank Conjecture holds for any non-trivial total permutation-invariant Boolean function. Moreover, we establish a relationship between the quantum/classical communication complexity and the approximate rank of permutation-invariant Boolean functions. This implies the correctness of the Log-approximate-rank Conjecture for permutation-invariant Boolean functions in both randomized and quantum settings (up to a logarithmic factor).
We study the one-way communication complexity of the f-Boolean Hidden Partition problem. Here, Alice is given an n-bit string and Bob is given Ω(n) disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of f on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where f is the parity function.
We establish classical and quantum communication bounds for f-Boolean Hidden Partition in terms of the sign degree d of f, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when d≤1 and quantum protocols when d≤2, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every f of sign degree d≥2, the randomized classical communication complexity of this problem is Θ(n1−1/d) while its quantum communication complexity lies between Ω(n1−2/d) and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions f admitting polynomial quantum advantage as those for which d≥2, with a new infinite family of exponential separations given by the case d=2.
This paper gives a nearly tight characterization of the quantum communication complexity of permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of permutation-invariant Boolean functions are quadratically equivalent (up to a polylogarithmic factor of the input size). Our results extend a recent line of research regarding query complexity to communication complexity, showing symmetry prevents exponential quantum speedups. Furthermore, we show that the Log-rank Conjecture holds for any non-trivial total permutation-invariant Boolean function. Moreover, we establish a relationship between the quantum/classical communication complexity and the approximate rank of permutation-invariant Boolean functions. This implies the correctness of the Log-approximate-rank Conjecture for permutation-invariant Boolean functions in both randomized and quantum settings (up to a polylogarithmic factor of the input size).
We study the advantages of quantum communication models over classical communication models that are equipped with a limited number of qubits of entanglement. In this direction, we give explicit partial functions on n bits for which reducing the entanglement increases the classical communication complexity exponentially. Our separations are as follows. For every k≥1: Q∥∗ versus R2∗: We show that quantum simultaneous protocols with Θ~(k5log3n) qubits of entanglement can exponentially outperform two-way randomized protocols with O(k) qubits of entanglement. This resolves an open problem from [Gav08] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [Gav19, GRT22]. R∥∗ versus Q∥∗: We show that classical simultaneous protocols with Θ~(klogn) qubits of entanglement can exponentially outperform quantum simultaneous protocols with O(k) qubits of entanglement, resolving an open question from [GKRW06, Gav19]. The best result prior to our work was a relational separation against protocols without entanglement [GKRW06]. R∥∗ versus R1∗: We show that classical simultaneous protocols with Θ~(klogn) qubits of entanglement can exponentially outperform randomized one-way protocols with O(k) qubits of entanglement. Prior to our work, only a relational separation was known [Gav08]. Our techniques can also be used to show advantages of quantum communication models over hybrid classical-quantum models, i.e., models that have a large amount of both classical communication and quantum simultaneous communication.
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 xj∈{0,1}n, and the referee must decide whether x1⊕⋯⊕xk=0n. For k≥8 and worst-case error at most 1/3, we prove quantum lower bounds of Ω(min{n,k2}) 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=k2 and worst-case error at most 1/3, we prove that quantum protocols require maximum message length 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 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(n1−1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error ε∈(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(logn), giving exponential separations in both regimes. The implicit constants are independent of k, and both quantum lower bounds become linear when k≥clogn for any fixed c>0.
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), 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∘AND2 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.