Cluster-State Witnesses of Finite-Speed Hidden Influences
Bell experiments rule out local common-cause explanations of quantum correlations, yet they do not exclude hidden influences that travel faster than light while still having a finite speed in a preferred frame. Multipartite spacetime arrangements turn this possibility into a constraint: two late parties that are outside each other's hidden-influence cones must remain Bell-local once the earlier events are fixed. Here, we formulate this constraint as a projected-polytope separation problem for cluster-state correlations, using only marginal data containing at most one late party. From linear cluster states, we construct a four-qubit witness with the bound $S_4\le 6$ and quantum value $4+2\sqrt2$, and a five-qubit witness with $S_5\le 10$ and quantum value $6+4\sqrt2$. We certify that the exposed faces are facets of the corresponding projected hidden-influence polytopes. These results identify linear-cluster graph states as certifiable and experimentally friendly resources for finite-speed hidden-influence tests.