Is the commuting local Hamiltonian problem in NP
In plain words
When all energy terms of a quantum system commute, deciding whether it has zero ground energy looks closer to a classical problem. It is proven to have short classical proofs in several special cases but not in general.
Precise statement
Given $H = \operatorname{sum}_i h_i$ with pairwise commuting k-local projectors $h_i$ on qudits of dimension $q$, decide whether the ground energy is 0 or at least $1/\operatorname{poly}(n)$. Is this in NP for all constant k and q? Known in NP: 2-local (Bravyi and Vyalyi 2003), some 3-local cases (Aharonov and Eldar 2011), 2D qubits (Schuch 2011), 2D qutrits and factorized terms of any dimension (Irani and Jiang, arXiv:2309.04910).
What would settle it
A classical witness and verifier for all constant k and q, or QMA-hardness (or hardness for a class not believed in NP) for some fixed k and q.