We know that the $2$-local Hamiltonian problem is (promise) QMA-complete, which under the reasonable assumption that BQP$\subsetneq$QMA implies that no fast quantum algorithm exists to determine the eigenstate, much less the energy, of the ground state of such a Hamiltonian.
An implication is that not even Nature herself can find the ground state of such a Hamiltonian, as it might take longer than the age of the universe to settle into such a state (or even to learn said energy).
But if Nature could never find the entangled ground state of our Hamiltonian, how much value is there in the ground state itself?
For example, would knowing the ground state of, say, some lump of metal for some hard, bespoke Hamiltonian help us in constructing a room-temperature superconductor, or help us achieve fusion or otherwise be of significant relevance?
This is contrasted with the classical case, wherein even knowing whether solutions to certain satisfiability problems exists would be of extreme industrial significance - even though we can similarly conclude, assuming that P$\subsetneq$NP and SETH etc., that a classical computer could take longer than the age of the universe to determine the existence of a solution.
Also, NP$\subseteq$QMA, so if our Hamiltonian just describes a classical NP problem then yes knowing the ground energy lets us solve the corresponding NP problem, but if the ground state were some highly-entangled eigenstate then is such a state still valuable if Nature cannot find it?
