We introduce Verifiable One-Time Programs (Ver-OTPs) and use them to construct single-round Open Secure Computation (OSC), a novel primitive enabling applications like (1) single-round sealed-bid auctions, (2) single-round and honest-majority atomic proposes -- a building block of consensus protocols, and (3) single-round differentially private statistical aggregation without pre-registration. First, we construct Ver-OTPs from single-qubit states and classical cryptographic primitives. Then, assuming a multi-key homomorphic scheme (MHE) with certain properties, we use Ver-OTPs with MHE to construct OSC. The underlying quantum requirement is minimal: only single-qubit states are needed alongside a hardware assumption on the receiver's quantum resources. Our work therefore provides a new framework for quantum-assisted cryptography that may be implementable with near-term quantum technology.