We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $\Pi$ and reject inputs that are $\varepsilon$-far from $\Pi$, while only probing a minuscule portion of their input.

Our algorithmic results include a general-purpose theorem that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable. Furthermore, we show quantum speedups for properties that lie outside of this family, such as graph bipartitneness.

We also investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity, thereby resolving a problem raised by Montanaro and de Wolf (Theory of Computing, 2016).