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In a wireless system with opportunistic spectrum sharing, secondary users equipped with cognitive radios attempt to access a radio spectrum that is not being used by the primary licensed users. On a given frequency channel, a secondary user can perform spectrum sensing to determine spatial or temporal opportunities for spectrum reuse. Whereas most prior works address either spatial or temporal sensing in isolation, we propose a joint spatial-temporal spectrum-sensing scheme that exploits information from spatial sensing to improve the performance of temporal sensing. We quantify the performance benefit of the joint spatial-temporal scheme over pure spatial sensing and pure temporal sensing based on counting-rule and linear quadratic (LQ) detectors. Finally, we analyze a multilevel quantization feedback scheme that can improve the performance of temporal sensing based on counting-rule detectors.