According to Bell's theorem, any physical theory based on local variables cannot fully reproduce quantum correlations. A critical question arises whether it is possible to develop an alternative framework based on nonlocal variables that can reproduce quantum correlations. Leggett proposed a nonlocal model, termed crypto-nonlocality, grounded in considerations of the reality of photon polarization. However, this model is inherently restricted to two dimensions. In this Letter, we extend the framework to higher dimensions and propose a method for constructing experimentally testable Leggett-type inequalities in arbitrary dimensions. From an information-theoretic perspective, our exploration of models that predict outcomes exceeding the scope of quantum mechanics is particularly intriguing and expected to deepen our understanding of quantum correlations.

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• Nonlocality
• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism