Similar to the class of classical NP-intermediate problems (e.g. Graph Isomorphism), is there any "QMA-intermediate" problem known, that is in QMA but not known to be QMA-complete? Has this been studied systematically before?
An example would be the computation of ground state energy of the Ising model with transverse magnetic fields, as described by [Cubitt+Montenaro-2013]. From the abstract:
In this work we characterise the complexity of [the calculation of ground-state energies of 2-body Hamiltonians on qubits with terms restricted to some fixed set S]. Depending on the subset S, the problem falls into one of the following categories: in P; NP-complete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete. The third of these classes contains NP and is contained within StoqMA.