The Impossibility of Efficient Quantum Weak Coin-Flipping

TL;DR

This paper proves that practical quantum weak coin-flipping protocols with very low bias require an exponentially large number of communication rounds, explaining their inherent difficulty.

Contribution

It establishes a new exponential lower bound on the rounds needed for quantum weak coin-flipping with small bias, improving previous bounds significantly.

Findings

Any protocol with bias ε requires at least exp(Ω(1/√ε)) rounds

Previous lower bound was only Ω(log log(1/ε))

Introduces the two-variable profile function for analysis

Abstract

How can two parties with competing interests carry out a fair coin flip, using only a noiseless quantum channel? This problem (quantum weak coin-flipping) was formalized more than 15 years ago, and, despite some phenomenal theoretical progress, practical quantum coin-flipping protocols with vanishing bias have proved hard to find. In the current work we show that there is a reason that practical weak quantum coin-flipping is difficult: any quantum weak coin-flipping protocol with bias $\epsilon$ must use at least $\exp ( \Omega (1/\sqrt{\epsilon} ))$ rounds of communication. This is a large improvement over the previous best known lower bound of $\Omega ( \log \log (1/\epsilon ))$ due to Ambainis from 2004. Our proof is based on a theoretical construction (the two-variable profile function) which may find further applications.