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A relational bipartite communication problem is presented that has an efficient quantum simultaneous-messages protocol, but no efficient classical two-way protocol. [ABSTRACT FROM AUTHOR]

Multiple players are each given one independent sample, about which they can only provide limited information to a central referee. Each player is allowed to describe its observed sample to the referee using a channel from a family of channels $\mathcal {W}$ , which can be instantiated to capture, among others, both the communication- and privacy-constrained settings. The referee uses the players’ messages to solve an inference problem on the unknown distribution that generated the samples. We derive lower bounds for the sample complexity of learning and testing discrete distributions in this information-constrained setting. Underlying our bounds is a characterization of the contraction in chi-square distance between the observed distributions of the samples when information constraints are placed. This contraction is captured in a local neighborhood in terms of chi-square and decoupled chi-square fluctuations of a given channel, two quantities we introduce. The former captures the average distance between distributions of channel output for two product distributions on the input, and the latter for a product distribution and a mixture of product distribution on the input. Our bounds are tight for both public- and private-coin protocols. Interestingly, the sample complexity of testing is order-wise higher when restricted to private-coin protocols. [ABSTRACT FROM AUTHOR]

A central server needs to perform statistical inference based on samples that are distributed over multiple users who can each send a message of limited length to the center. We study problems of distribution learning and identity testing in this distributed inference setting and examine the role of shared randomness as a resource. We propose a general-purpose simulate-and-infer strategy that uses only private-coin communication protocols and is sample-optimal for distribution learning. This general strategy turns out to be sample-optimal even for distribution testing among private-coin protocols. Interestingly, we propose a public-coin protocol that outperforms simulate-and-infer for distribution testing and is, in fact, sample-optimal. Underlying our public-coin protocol is a random hash that when applied to the samples minimally contracts the chi-squared distance of their distribution to the uniform distribution. [ABSTRACT FROM AUTHOR]

This paper addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication “layouts”: two-way interactive communication, one-way communication and simultaneous message passing (SMP). I demonstrate a functional problem $ { \widetilde {cEq}_{T}}$ , whose communication complexity is $O\left ({(\log n)^{2}}\right)$ in the quantum version of the SMP and $\tilde \Omega \left ({ \sqrt {n}}\right)$ in the classical (randomized) version of SMP. Second, this paper contributes to understanding the power of the weakest commonly studied regime of quantum communication–SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model “as weak as possible”). Our function $ { \widetilde {cEq}_{T}}$ has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness. [ABSTRACT FROM AUTHOR]