Indeed for the standard, conventional QMA-complete Local Hamiltonian problem,
- An all-powerful Merlin and a BQP-capable Arthur agree on a local Hamiltonian $\mathcal H$ acting on $n$ qubits;
- Merlin provides Arthur the $n$-qubit witness $|\psi\rangle$ that is promised to be in an eigenstate of $\mathcal H$ having an eigenvalue less than some value $a$;
- Arthur uses Hamiltonian simulation on $\mathcal H$ to run a quantum phase estimation algorithm on $|\psi\rangle$; and
- Arthur can validate whether the measured energy eigenvalue is less than $a$.
As @Norbert indicated the verifier Arthur must be capable of running the BQP phase estimation algorithm (including the Hamiltonian simulation), which means Arthur should be quantum (assuming that BPP$\ne$BQP).
But, following a breakthrough by Mahadev, a classical Arthur can use post-quantum cryptography to outsource the running of BQP algorithms to a quantum machine! See this somewhat more accessible Quanta magazine article on Mahadev's results.
As @Norbert also indicates in the comments, however, it's not entirely clear if Mahadev's protocol carries through for the verification of QMA certificates. For example, an all-powerful QMA machine Merlin may provide a certificate $|\psi\rangle$ to a BQP machine, Guinevere, who then engages in a post-quantum cryptographic protocol with the fully classical Arthur. But, Merlin may be sneaky enough to trap Guinevere into being able to convince Arthur that Guinevere has received the proper quantum certificate $|\psi\rangle$, when in reality she's received garbage $|\psi'\rangle$. (I think the results do carry over, just thinking about the QMA certificate for group non-membership and the two test that Arthur can run. But, it's not clear and maybe a separate question).
However, Mahadev's protocol likely does carry over to verifications of QCMA certificates. There an powerful Merlin provides Arthur with a classical description of a quantum circuit that can be used to build the witness string $|\psi\rangle$, from which then Arthur could instruct Guinevere to construct and execute the verification. In the QCMA case, all-powerful Merlin never directly engages with BQP-bounded Guinevere.