The class QMA plays a fundamental role in quantum complexity theory and it has found surprising connections to condensed matter physics and in particular in the study of the minimum energy of quantum systems. In this paper, we further investigate the class QMA and its related class QCMA by asking what makes quantum witnesses potentially more powerful than classical ones. We provide a definition of a new class, SQMA, where we restrict the possible quantum witnesses to the "simpler" subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Surprisingly, we prove that this class is equal to QMA, hence providing a new characterisation of the class QMA. We also prove the analogous result for QMA(2) and describe a new complete problem for QMA and a stronger lower bound for the class QMA$_1$.