Exponential Quantum-Classical Gaps in Multiparty Nondeterministic Communication Complexity

TL;DR

This paper demonstrates exponential gaps between different types of quantum nondeterministic communication complexities in multiparty settings, revealing fundamental separations and implications for complexity measures.

Contribution

It establishes exponential separations between multiparty qp- and cma-communication complexities, introducing new lower bounds and techniques.

Findings

Multiparty qp-communication can be exponentially stronger than cma-communication.

Existence of a total function hard for cma but easy for qp.

Nondeterministic rank can be exponentially lower than discrepancy.

Abstract

There are three different types of nondeterminism in quantum communication: i) $\nqp$-communication, ii) $\qma$-communication, and iii) $\qcma$-communication. In this \redout{paper} we show that multiparty $\nqp$-communication can be exponentially stronger than $\qcma$-communication. This also implies an exponential separation with respect to classical multiparty nondeterministic communication complexity. We argue that there exists a total function that is hard for $\qcma$-communication and easy for $\nqp$-communication. The proof of it involves an application of the pattern tensor method and a new lower bound for polynomial threshold degree. Another important consequence of this result is that nondeterministic rank can be exponentially lower than the discrepancy bound.