We give the first exponential quantum advantage in the general interactive three-party Numbers-on-Forehead (NOF) model for a decision problem. Previous separations hold only for restricted protocols like one-way communication for a relation. We construct an explicit partial Boolean function, the Interleaved Unitary Product problem, that requires only $O(\log n)$ NOF quantum communication but $\widetildeΩ(n^{1/32})$ randomized communication. This function builds on the two-party unitary product problem of Arunachalam, Girish, and Lifshitz (TQC 2024).
The main technical obstacle is that discrepancy, the standard lower-bound method for NOF, also lower-bounds quantum communication. We instead develop a regularity-based argument for randomized NOF lower bounds, building on the approach of Kelley, Lovett, and Meka and adapting the regularity decomposition of Abboud, Fischer, Kelley, Lovett, and Meka (STOC 2024) to cylinder intersections. Combined with matrix-product estimates of Arunachalam, Girish, and Lifshitz, this yields our randomized lower bound.
We give the first exponential quantum advantage in the general interactive three-party Numbers-on-Forehead (NOF) model for a decision problem. Previous separations hold only for restricted protocols like one-way communication for a relation. We construct an explicit partial Boolean function, the Interleaved Unitary Product problem, that requires only O(logn) NOF quantum communication but Ω(n1/32) randomized communication. This function builds on the two-party unitary product problem of Arunachalam, Girish, and Lifshitz (TQC 2024).
The main technical obstacle is that discrepancy, the standard lower-bound method for NOF, also lower-bounds quantum communication. We instead develop a regularity-based argument for randomized NOF lower bounds, building on the approach of Kelley, Lovett, and Meka and adapting the regularity decomposition of Abboud, Fischer, Kelley, Lovett, and Meka (STOC 2024) to cylinder intersections. Combined with matrix-product estimates of Arunachalam, Girish, and Lifshitz, this yields our randomized lower bound.
Numbers-on-Forehead (NOF) communication model is a central model in communication complexity. As a restricted variant, one-way NOF model is of particular interest. Establishing strong one-way NOF lower bounds would imply circuit lower bounds, resolve well-known problems in additive combinatorics, and yield wide-ranging applications in areas such as cryptography and distributed computing. However, proving strong lower bounds in one-way NOF communication remains highly challenging; many fundamental questions in one-way NOF communication remain wide open. One of the fundamental questions, proposed by Gavinsky and Pudlák (CCC 2008), is to establish an explicit exponential separation between quantum and classical one-way NOF communication.
In this paper, we resolve this open problem by establishing the first exponential separation between quantum and randomized communication complexity in one-way NOF model. Specifically, we define a lifted variant of the Hidden Matching problem of Bar-Yossef, Jayram, and Kerenidis (STOC 2004) and show that it admits an (O(logn))-cost quantum protocol in the one-way NOF setting. By contrast, we prove that any k-party one-way randomized protocol for this problem requires communication Ω(2k/3n1/3). Notably, our separation applies even to a generalization of k-player one-way communication, where the first player speaks once, and all other k−1 players can communicate freely.
We prove a lifting theorem from two-party public-coin one-way communication to multiparty public-coin one-way number-on-forehead (NOF) communication. For every fixed k≥2 and prime q>2k, there is a generalized inner product gadget $\GIP_{q,r}^k:(\F_q^r)^k\to\F_q$ with r=Ok(q/logq) such that, for every partial Boolean function $f:D\to\bits$, where $D\subseteq\F_q\times\F_q$, \[ R_{1/3}^1(f)-O(1) \le R_{1/6}^{1,\NOF}\bigl(f\circ\GIP_{q,r}^k\bigr) \le R_{1/6}^1(f). \] Thus, composition with the gadget preserves one-way randomized communication complexity up to an additive constant and a change in the error parameter. The lower bound holds in the general one-way NOF model, where the last player sees the entire gadget input. This extends the deterministic one-way NOF lifting theorem of Yang and Zhang to randomized protocols, and extends the randomized lifting result of Wang and Wu from the conservative model to the general one-way NOF model. Our proof introduces a one-way cylinder partition bound that lower bounds public-coin one-way NOF communication complexity. We show that, for the lifted function, this bound is at least half the one-way partition bound of the outer function. The main technical step transfers a dual solution between the two bounds, using Möbius inversion and a discrepancy estimate for generalized inner product to control the loss. Combining this transfer with the characterization of two-party one-way randomized communication complexity by the one-way partition bound yields the lifting theorem.
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.
Quantum versus classical separation plays a central role in understanding the advantages of quantum computation. In this paper, we present the first exponential separation between quantum and bounded-error randomized communication complexity in a variant of the Number-on-Forehead (NOF) model. Namely, the three-player Simultaneous Number-on-Forehead model. Specifically, we introduce the Gadgeted Hidden Matching Problem and show that it can be solved using only O(logn) simultaneous quantum communication. In contrast, any simultaneous randomized protocol requires Ω(n1/16) communication. On the technical side, a key obstacle in separating quantum and classical communication in NOF models is that all known randomized NOF lower bound tools, such as the discrepancy method, typically apply to both randomized and quantum protocols. In this regard, our technique provides a new method for proving randomized lower bounds in the NOF setting and may be of independent interest beyond the separation result.
We consider lifting theorems that transfer lower bounds for two-party communication problems to multiparty communication problems. In particular, following the deterministic Number-on-Forehead (NOF) lifting framework of Yang and Zhang, we study randomized and quantum one-way NOF lifting for composed problems F(z,𝐱) = f(z,G(𝐱)). We work in a one-way NOF model in which only the last player’s view is restricted. The other players have their usual NOF views and communicate as usual, but the last player sees only the gadget output G(𝐱), not the gadget input 𝐱. This kind of restricted-view has appeared in the NOF literature under the name conservative model. Our main contribution is a pair of lifting theorems for this model. In the randomized setting, we show that lifting follows when each preimage G^{-1}(v), the set of gadget inputs with output v, looks pseudorandom to large cylinder intersections. In the quantum setting, we prove an analogous theorem. Equivalently, conservative protocols for F(z,𝐱) = f(z,G(𝐱)) can be converted into two-party one-way protocols for f(z,v) with comparable cost, and the error loss controlled by the corresponding pseudorandomness parameters. Thus, this restriction isolates a setting in which both randomized and quantum one-way NOF lifting can be proved by a direct simulation argument. We prove the required pseudorandomness properties for the generalized inner product gadget over finite fields, using the multiparty character-sum bounds of Yang and Zhang, and for random gadgets, which give non-explicit lifting. As applications, Boolean Hidden Matching yields a randomized-versus-quantum separation in the conservative NOF model, and lifting INDEX gives randomized and quantum conservative NOF lower bounds for O(log n) players.
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].