We introduce the Quantum Operating System (QOS), a unified system stack for managing quantum resources while mitigating their inherent limitations, namely their limited and noisy qubits, (temporal and spatial) heterogeneities, and load imbalance. QOS features the $\textit{QOS compiler}$ -- a modular and composable compiler for analyzing and optimizing quantum applications to run on small and noisy quantum devices with high performance and configurable overheads. For scalable execution of the optimized applications, we propose the $\textit{QOS runtime}$ -- an efficient quantum resource management system that multi-programs and schedules the applications across space and time while achieving high system utilization, low waiting times, and high-quality results. We evaluate QOS on real quantum devices hosted by IBM, using 7000 real quantum runs of more than 70.000 benchmark instances. We show that the QOS compiler achieves 2.6--456.5$\times$ higher quality results, while the QOS runtime further improves the quality by 1.15--9.6$\times$ and reduces the waiting times by up to 5$\times$ while sacrificing only 1--3\% of results quality (or fidelity).

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