Can local measurements reveal a quantum material's global topology?

In topological quantum materials, a global property called Berry phase encodes essential information about the ground state. Using cumulant analysis, this work shows that knowing all correlations between particles up to order N−1 generically fails to determine the Berry phase in an N-particle system. In the thermodynamic limit, no finite set of local measurements suffices. Exceptions exist: free-electron systems and symmetry-constrained materials can reveal topology locally, but most interacting systems cannot.