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Quantum computers face challenges due to hardware constraints, noise errors, and heterogeneity, and face fundamental design tradeoffs between key performance metrics such as quantum fidelity and system utilization. This substantially complicates managing quantum resources to scale the size and number of quantum algorithms that can be executed reliably in a given time. We introduce QOS, a modular quantum operating system that holistically addresses the challenges of quantum resource management by systematically exploring key design tradeoffs across the stack.QOS exposes a hardware-agnostic API for transparent quantum job execution, mitigates hardware errors, and systematically multi-programs and schedules the jobs across space and time to achieve high quantum fidelity in a resource-efficient manner. QOS's modular design enables synergistic cross- and intra-layer optimizations, while introducing new concepts such as compatibility-based multi-programming and effective utilization. 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 achieves 2.6--456.5× higher fidelity, increases resource utilization by up to 9.6×, and reduces waiting times by up to 5× while sacrificing only 1--3% fidelity, on average, compared to the baselines.