Indefinite causal order (ICO), a striking feature of quantum mechanics, has attracted broad theoretical and experimental interest. While ICO holds promise for quantum information processing, akin to quantum entanglement, its experimental realization and rigorous verification remain fundamental challenges. Here, we demonstrate and certify generalized multidimensional multipartite ICOs using a programmable integrated-photonic quantum chip implementing a generalized quantum switch. By violating definite causal order bounds with high statistical significance, we experimentally characterize and verify these generalized ICOs. Furthermore, we explore a resource theory of causal nonseparability by implementing a single-copy distillation protocol. Our Letter constitutes a major step toward a generalized framework for quantum causality.

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• Quantum correlations, foundations & formalism