Determining whether a quantum state set admits a classical explanation is a fundamental problem in quantum information theory, as it reveals whether the information encoded in state preparation can be reproduced by classical means. Here we study classically simulable state sets, namely those admitting realizations using shared randomness and preparation devices that individually generate only pairwise commuting state sets. We develop a complete semidefinite programming (SDP) hierarchy to characterize the set of classically simulable state sets. The key step is a reformulation of classical simulability as a feasibility problem involving deterministic response functions and auxiliary positive-operator-valued measures (POVMs) simulable by rank-one projective measurements. We then establish a complete SDP hierarchy for rank-one projectively simulable POVMs, which in turn induces computable outer approximations converging to the set of classically simulable state sets. The corresponding dual SDP further yields affine witnesses that can be used to rule out classical simulability. Applying the hierarchy to state sets under depolarizing noise yields computable upper bounds on the critical visibility. For several highly symmetric examples, these bounds are matched by explicit classical simulations, providing evidence for the practical usefulness of the hierarchy.