Exponential Separation of Quantum and Classical One-Way Numbers-on-Forehead Communication
This addresses a fundamental question in communication complexity with implications for circuit lower bounds, additive combinatorics, cryptography, and distributed computing.
The paper resolves the open problem of establishing an explicit exponential separation between quantum and classical one-way NOF communication by showing that a lifted variant of the Hidden Matching problem has an O(log n)-cost quantum protocol, while any k-party one-way randomized protocol requires Ω(n^{1/3}/2^{k/3}) communication.
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.