Physical theories constrained with local quantum structure and satisfying the no-signaling principle can allow beyond-quantum global states. In a standard Bell experiment, correlations obtained from any such beyond-quantum bipartite state can always be reproduced by quantum states and measurements, suggesting the local quantum structure and no-signaling to be the axioms to isolate quantum correlations. In this Letter, however, we show that if the Bell experiment is generalized to allow local quantum inputs, then beyond-quantum correlations can be generated by every beyond-quantum state. This gives us a way to certify the beyond quantumness of locally quantum no-signaling theories and in turn suggests the requirement of additional information principles along with the local quantum structure and no-signaling principle to isolate quantum correlations. More importantly, our work establishes that the additional principle(s) must be sensitive to the quantum signature of local inputs. We also generalize our results to multipartite locally quantum no-signaling theories and further analyze some interesting implications.

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• Nonlocality
• Quantum correlations in quantum information
• Quantum entanglement
• Quantum foundations
• Quantum measurements