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We study 'Merlinized' versions of the recently defined Guided LocalHamiltonian problem, which we call 'Guidable Local Hamiltonian' problems.Unlike their guided counterparts, these problems do not have a guiding stateprovided as a part of the input, but merely come with the promise that oneexists. We consider in particular two classes of guiding states: those that canbe prepared efficiently by a quantum circuit; and those belonging to a class ofquantum states we call classically evaluatable, for which it is possible toefficiently compute expectation values of local observables classically. Weshow that guidable local Hamiltonian problems for both classes of guidingstates are $\mathsf{QCMA}$-complete in the inverse-polynomial precisionsetting, but lie within $\mathsf{NP}$ (or $\mathsf{NqP}$) in the constantprecision 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 asquantum Ans\"atze prepared by quantum heuristics, as long as one has access toquantum phase estimation. In relation to the quantum PCP conjecture, we (i)define a complexity class capturing quantum-classical probabilisticallycheckable proof systems and show that it is contained in$\mathsf{BQP}^{\mathsf{NP}[1]}$ for constant proof queries; (ii) give a no-goresult on 'dequantizing' the known quantum reduction which maps a$\mathsf{QPCP}$-verification circuit to a local Hamiltonian with constantpromise gap; (iii) give several no-go results for the existence of quantum gapamplification procedures that preserve certain ground state properties; and(iv) propose two conjectures that can be viewed as stronger versions of theNLTS theorem. Finally, we show that many of our results can be directlymodified to obtain similar results for the class $\mathsf{MA}$.

Guidable Local Hamiltonian Problems with Implications to Heuristic Ans\"atze State Preparation and the Quantum PCP Conjecture