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.
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.
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 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.
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.
For every fixed k >= 2 and prime q > 2k, this paper proves that generalized inner product of dimension r = O_k(sqrt(q)/log(q)) lifts the public-coin one-way randomized communication complexity of every partial Boolean function on F_q x F_q to the general one-way number-on-forehead model, up to an additive constant and error parameters 1/6 and 1/3. This improves the sufficient O_k(q/log(q)) dimension in Wang's preceding general randomized lifting theorem. The proof uses a localized polynomial approximation on a cone, controlled coefficient norm, and an error-parameter gap to absorb local dual mass. The manuscript includes complete definitions, the approximation and transfer proofs, exact integer gadget parameters, source comparisons, limitations, and an AI-assistance disclosure. No necessary-dimension lower bound or constant-dimension lifting result is claimed. The record contains the manuscript PDF, standalone LaTeX file, and reproducible source archive.
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.