Exponential Separation of Quantum and Classical One-Way Numbers-on-Forehead Communication

• URL: http://arxiv.org/abs/2603.22795v1
• Date: Tue, 24 Mar 2026 04:38:22 GMT
• Title: Exponential Separation of Quantum and Classical One-Way Numbers-on-Forehead Communication
• Abstract summary: Numbers-on-Forehead (NOF) communication model is a central model in communication complexity.<n>In this paper, we establish the first exponential separation between quantum and randomized communication complexity in one-way NOF model.
• Score: 14.595648710226735
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 ($O(\log n)$)-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 $Ω(\frac{n^{1/3}}{2^{k/3}})$. 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.

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