The complexity class NP has a natural definition that has been widely studied in various contexts in classical complexity theory. For the quantum version of NP, however, one can make several choices while defining a class of problems that can be verified in quantum polynomial time. QCMA, QMA and QMA(2) are examples of these classes. Although these three classes have been studied extensively in the past two decades, little is known about their comparative power. Here we study the relation between these classes using different techniques: either through oracle separations or by studying restricted variants of these classes. First, to study the power of classical proofs, we examine how different design choices for an oracle influence the complexity of quantum property testing problems that are defined with respect to that oracle. Specifically, we represent a regular graph with even degree as an invertible function f, and explore various oracle models for accessing f. First, we introduce a one-query QMA protocol to test whether the graph encoded by f contains a small disconnected component. Then, using representation theory, we demonstrate that a classical witness cannot assist a quantum verifier in efficiently solving this problem when using an in-place oracle. Interestingly, a minor alteration to the conventional oracle setup can prevent even a quantum verifier with unlimited witness access from solving the problem efficiently. Second, in an attempt to understand the power of QMA(2), we explore a direction proposed by Jeronimo and Wu -- where it was shown that QMA(2) with non-negative amplitudes has the ability to solve NEXP-hard problems, and hence amplification for that class implies QMA(2) = NEXP. To give negative evidence for this direction, we explore QMA with non-negative amplitudes. When only the completeness condition is altered, this modified class remains equivalent to standard QMA. However, if both completeness and soundness are changed, the resulting class becomes significantly more powerful. We demonstrate that QMA+ with one fixed completeness-soundness gap equals NEXP, while with a different constant gap, it remains equivalent to QMA. This result limits the possibility of proving QMA(2) = NEXP by amplification, since any amplification technique must fail for QMA+. Lastly, we use first these techniques to establish that QMA, when the verifier is allowed to perform a single non-collapsing measurement, is equivalent to NEXP--answering an open question posed by Aaronson. Central to several findings influenced by Blier and Tapp is a non-physical property testing problem, which involves determining whether a quantum state is close to a specific basis element. This viewpoint leads us to a variation of QMA in which a single quantum proof is strictly less powerful than two unentangled proofs—under the assumption that EXP /= NEXP. This provides a novel path towards proving that QMA(2)=NEXP, while avoiding a key limitation of the previous proposed method. In our modification, each proof is required to exhibit a type of multipartite unentanglement: specifically, once one register is traced out, a small subset of qubits remains separable from the rest of the quantum state.