Necessary and Sufficient Condition for Randomness Certification from Incompatibility

Quantum randomness can be certified from probabilistic behavior demonstrating Bell nonlocality or Einstein-Podolsky-Rosen steering, leveraging outcomes from uncharacterized devices. However, in standard spot-checking protocols, such nonlocal correlations are not always sufficient for this task, necessitating the identification of required minimum quantum resources. In this Letter, we focus on the bipartite scenario and provide the necessary and sufficient condition for nonzero certified randomness under any arbitrary but fixed input, formulated in terms of measurement incompatibility. Further, we develop practical approaches to detect it. Firstly, we show that the steering-based randomness can be certified if and only if the correlations arise from a measurement compatibility structure that is not isomorphic to a hypergraph containing a star subgraph. In such a structure, the central measurement is individually compatible with the measurements at branch sites, ruling out the possibility of certified randomness in the central measurement outcomes. Subsequently, we generalize this result to the Bell scenario, proving that the violation of any chained Bell inequality involving a finite number of inputs and outputs excludes such a compatibility structure, thereby validating all chained inequalities as credible witnesses for randomness certification. Our results point out the role of an incompatibility structure in generating random numbers, offering a way to identify minimum quantum resources for the task.