  We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as
the class of problems that are efficiently verifiable given alternating quantum
proofs that may be entangled with each other. We prove $\mathsf{QEPH}$
collapses to its second level. In fact, we show that a polynomial number of
alternations collapses to just two. As a consequence, $\mathsf{QEPH} =
\mathsf{QRG(1)}$, the class of problems having one-turn quantum refereed games,
which is known to be contained in $\mathsf{PSPACE}$. This is in contrast to the
unentangled quantum polynomial hierarchy $\mathsf{QPH}$, which contains
$\mathsf{QMA(2)}$.
  We also introduce a generalization of the quantum-classical polynomial
hierarchy $\mathsf{QCPH}$ where the provers send probability distributions over
strings (instead of strings) and denote it by $\mathsf{DistributionQCPH}$.
Conceptually, this class is intermediate between $\mathsf{QCPH}$ and
$\mathsf{QPH}$. We prove $\mathsf{DistributionQCPH} = \mathsf{QCPH}$,
suggesting that only quantum superposition (not classical probability)
increases the computational power of these hierarchies. To prove this equality,
we generalize a game-theoretic result of Lipton and Young (1994) which says
that the provers can send distributions that are uniform over a polynomial-size
support. We also prove the analogous result for the polynomial hierarchy, i.e.,
$\mathsf{DistributionPH} = \mathsf{PH}$. These results also rule out certain
approaches for showing $\mathsf{QPH}$ collapses.
  Finally, we show that $\mathsf{PH}$ and $\mathsf{QCPH}$ are contained in
$\mathsf{QPH}$, resolving an open question of Gharibian et al. (2022).
