  We initiate the study of quantum races, games where two or more quantum
computers compete to solve a computational problem. While the problem of
dueling algorithms has been studied for classical deterministic algorithms, the
quantum case presents additional sources of uncertainty for the players. The
foremost among these is that players do not know if they have solved the
problem until they measure their quantum state. This question of `when to
measure?' presents a very interesting strategic problem. We develop a
game-theoretic model of a multiplayer quantum race, and find an approximate
Nash equilibrium where all players play the same strategy. In the two-party
case, we further show that this strategy is nearly optimal in terms of payoff
among all symmetric Nash equilibria. A key role in our analysis of quantum
races is played by a more tractable version of the game where there is no
payout on a tie; for such races we completely characterize the Nash equilibria
in the two-party case.
  One application of our results is to the stability of the Bitcoin protocol
when mining is done by quantum computers. Bitcoin mining is a race to solve a
computational search problem, with the winner gaining the right to create a new
block. Our results inform the strategies that eventual quantum miners should
use, and also indicate that the collision probability---the probability that
two miners find a new block at the same time---would not be too high in the
case of quantum miners. Such collisions are undesirable as they lead to forking
of the Bitcoin blockchain.
