  We study 'Merlinized' versions of the recently defined Guided Local
Hamiltonian problem, which we call 'Guidable Local Hamiltonian' problems.
Unlike their guided counterparts, these problems do not have a guiding state
provided as a part of the input, but merely come with the promise that one
exists. We consider in particular two classes of guiding states: those that can
be prepared efficiently by a quantum circuit; and those belonging to a class of
quantum states we call classically evaluatable, for which it is possible to
efficiently compute expectation values of local observables classically. We
show that guidable local Hamiltonian problems for both classes of guiding
states are $\mathsf{QCMA}$-complete in the inverse-polynomial precision
setting, but lie within $\mathsf{NP}$ (or $\mathsf{NqP}$) in the constant
precision regime when the guiding state is classically evaluatable.
  Our completeness results show that, from a complexity-theoretic perspective,
classical Ans\"atze selected by classical heuristics are just as powerful as
quantum Ans\"atze prepared by quantum heuristics, as long as one has access to
quantum phase estimation. In relation to the quantum PCP conjecture, we (i)
define a complexity class capturing quantum-classical probabilistically
checkable proof systems and show that it is contained in
$\mathsf{BQP}^{\mathsf{NP}[1]}$ for constant proof queries; (ii) give a no-go
result on 'dequantizing' the known quantum reduction which maps a
$\mathsf{QPCP}$-verification circuit to a local Hamiltonian with constant
promise gap; (iii) give several no-go results for the existence of quantum gap
amplification procedures that preserve certain ground state properties; and
(iv) propose two conjectures that can be viewed as stronger versions of the
NLTS theorem. Finally, we show that many of our results can be directly
modified to obtain similar results for the class $\mathsf{MA}$.
