Area laws describe how entanglement entropy scales and thus provide important necessary conditions for efficient quantum many-body simulation, but they do not, by themselves, yield a direct measure of computational complexity. Here we introduce a complementary framework based on $p$-particle positivity conditions from reduced density matrix (RDM) theory. These conditions form a hierarchy of $N$-representability constraints for an RDM to correspond to a valid $N$-particle quantum system. We prove a general complexity bound: if a quantum system is solvable with level-$p$ positivity independent of its size, then its entanglement complexity scales polynomially with order $p$. We also show that locality provides one mechanism for satisfying this complexity bound, establishing connections to area laws. Positivity laws provide a rigorous framework for understanding entanglement complexity and certifying when many-body methods—including RDM approaches—can efficiently simulate correlated quantum matter and materials.

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• Computational complexity
• Electronic structure
• Quantum entanglement
• Density matrix methods
• Many-body techniques