Experimental violation of a Bell-like inequality for causal order

Quantum mechanics is compatible with scenarios where physical processes happen in an indefinite order. In theory, this feature could be detected through violations of inequalities on the observed correlations, analogous to Bell inequalities. However, experimental demonstrations of such violations have been missing until recently due to the complexity of the required setup. Here, we report an experimental violation of a Bell-like inequality involving the correlations of four parties, one of which is spacelike separated from the others. Our demonstration uses 3-kilometer fiber spools to simulate spacelike separation and achieves high-speed operations in photonic time-bin encoding, nanosecond synchronization, and accurate temperature stabilization. These experimental advances enable a violation by 5.7 SDs and open a path toward a certification of indefinite order in conditions that guarantee spacelike separation with existing state-of-the-art devices. However, the certification is not device independent, as it relies on knowledge about the setup to exclude bidirectional signaling-a loophole inherent to implementations in classical acyclic spacetimes, which may be resolved in future quantum-spacetime tests.