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We show that bipartite Bell inequalities based on the Einstein–Podolsky–Rosen criterion for elements of reality and derived from the properties of some hyper-entangled states: (a) Allow feasible experimental verifications of the fact that the impossibility of elements of reality grows exponentially with the size of the subsystems, and (b) significantly reduces the minimum detection efficiency required to experimentally refute elements of reality without the fair sampling assumption.
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