  The polynomial-time hierarchy ($\mathrm{PH}$) has proven to be a powerful
tool for providing separations in computational complexity theory (modulo
standard conjectures such as $\mathrm{PH}$ does not collapse). Here, we study
whether two quantum generalizations of $\mathrm{PH}$ can similarly prove
separations in the quantum setting. The first generalization, $\mathrm{QCPH}$,
uses classical proofs, and the second, $\mathrm{QPH}$, uses quantum proofs. For
the former, we show quantum variants of the Karp-Lipton theorem and Toda's
theorem. For the latter, we place its third level, $\mathrm{Q} \Sigma_3$, into
$\mathrm{NEXP}$ {using the Ellipsoid Method for efficiently solving
semidefinite programs}. These results yield two implications for
$\mathrm{QMA}(2)$, the variant of Quantum Merlin-Arthur ($\mathrm{QMA}$) with
two unentangled proofs, a complexity class whose characterization has proven
difficult. First, if $\mathrm{QCPH} = \mathrm{QPH}$ (i.e., alternating
quantifiers are sufficiently powerful so as to make classical and quantum
proofs "equivalent"), then $\mathrm{QMA}(2)$ is in the Counting Hierarchy
(specifically, in $\mathrm{P}^{\mathrm{PP}^{\mathrm{PP}}}$). Second, unless
$\mathrm{QMA}(2)={\mathrm{Q} \Sigma_3}$ (i.e., alternating quantifiers do not
help in the presence of "unentanglement"), $\mathrm{QMA}(2)$ is strictly
contained in $\mathrm{NEXP}$.
