The class of decision problems for which a "yes" answer can be verified by a public-coin quantum AM protocol, as follows. Arthur generates a uniformly random (classical) string and sends it to Merlin. Merlin responds with a polynomial-size quantum certificate, on which Arthur can perform any BQP operation. The completeness and soundness requirements are the same as for AM.
Defined by Marriott and Watrous [MW05].
Contains QMA and is contained in QIP[2] and BP•PP (and therefore PSPACE).
Same as QAM except Arthur sends Merlin polynomially many halves of EPR-pairs instead of random coinflips.
Trivially contains QMA and NIQSZK, and is contained in QIP[2].
Defined in [KGN19], where it was shown that estimating the maximum entropy and minimum trace distance to the maximally mixed state among all states in the image of a quantum channel are complete problems.
In [GK25] it was shown that determining if a local Hamiltonian has a low energy state with high entanglement entropy is complete for qq-QAM. It was also shown that additive approximation to the free energy (equivalently multiplicative approximation to the quantum partition function) can be computed in qq-QAM.