Complexity limitations on one-turn quantum refereed games

This paper studies complexity theoretic aspects of quantum refereed games, which are abstract games between two competing players that send quantum states to a referee, who performs an efficiently implementable joint measurement on the two states to determine which of the player wins. The complexity class $\mathrm{QRG}(1)$ contains those decision problems for which one of the players can always win with high probability on yes-instances and the other player can always win with high probability on no-instances, regardless of the opposing player's strategy. This class trivially contains $\mathrm{QMA} \cup \text{co-}\mathrm{QMA}$ and is known to be contained in $\mathrm{PSPACE}$. We prove stronger containments on two restricted variants of this class. Specifically, if one of the players is limited to sending a classical (probabilistic) state rather than a quantum state, the resulting complexity class $\mathrm{CQRG}(1)$ is contained in $\exists\cdot\mathrm{PP}$ (the nondeterministic polynomial-time operator applied to $\mathrm{PP}$); while if both players send quantum states but the referee is forced to measure one of the states first, and incorporates the classical outcome of this measurement into a measurement of the second state, the resulting class $\mathrm{MQRG}(1)$ is contained in $\mathrm{P}\cdot\mathrm{PP}$ (the unbounded-error probabilistic polynomial-time operator applied to $\mathrm{PP}$).