  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 $\mathrm{PP}$).
