  Testing the symmetries of quantum states and channels provides a way to
assess their usefulness for different physical, computational, and
communication tasks. Here, we establish several complexity-theoretic results
that classify the difficulty of symmetry-testing problems involving a unitary
representation of a group and a state or a channel that is being tested. In
particular, we prove that various such symmetry-testing problems are complete
for BQP, QMA, QSZK, QIP(2), QIP_EB(2), and QIP, thus spanning the prominent
classes of the quantum interactive proof hierarchy and forging a non-trivial
connection between symmetry and quantum computational complexity. Finally, we
prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and
QAM, while leaving it as an intriguing open question to determine whether these
problems are complete for these classes.
