Quantum Merlin-Arthur proof systems are believed to be stronger than both their classical counterparts and “stand-alone” quantum computers when Arthur is assumed to operate in \(\varOmega (\log n)\) space. No hint of such an advantage over classical computation had emerged from research on smaller space bounds, which had so far concentrated on constant-space verifiers. We initiate the study of quantum Merlin-Arthur systems with space bounds in \(\omega (1) \cap o(\log n)\) , and exhibit a problem family \(\mathcal {F}\) , whose yes-instances have proofs that are verifiable by polynomial-time quantum Turing machines operating in this regime. We show that no problem in \(\mathcal {F}\) has proofs that can be verified classically or is solvable by a stand-alone quantum machine in polynomial time if standard complexity assumptions hold. Unlike previous examples of small-space verifiers, our protocols require only subpolynomial-length quantum proofs.