# Quantum coin hedging, and a counter measure

## TL;DR

This paper explores quantum hedging in cryptographic protocols, demonstrating its occurrence in quantum coin flipping and providing formulas for sequential two-outcome games, highlighting potential security challenges.

## Contribution

It shows that perfect quantum hedging occurs in quantum coin flipping protocols and provides a formula for sequential two-outcome board games, extending understanding of hedging limitations.

## Key findings

- Quantum hedging occurs in quantum coin flipping.
- Hedging cannot occur in sequential two-outcome board games.
- A formula for the value of sequential two-outcome board games is derived.

## Abstract

A quantum board game is a multi-round protocol between a single quantum player against the quantum board. Molina and Watrous discovered quantum hedging. They gave an example for perfect quantum hedging: a board game with winning probability < 1, such that the player can win with certainty at least 1-out-of-2 quantum board games played in parallel. Here we show that perfect quantum hedging occurs in a cryptographic protocol - quantum coin flipping. For this reason, when cryptographic protocols are composed, hedging may introduce serious challenges into their analysis. We also show that hedging cannot occur when playing two-outcome board games in sequence. This is done by showing a formula for the value of sequential two-outcome board games, which depends only on the optimal value of a single board game; this formula applies in a more general setting, in which hedging is only a special case.