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In quantum lattice systems, we prove that any stationary state with power-law(or even exponential) decay of spatial correlations has vanishing macroscopictemporal order in the thermodynamic limit. Assuming translational invariance,we obtain a similar bound on the temporal order between local operators at latetimes. Our proofs do not require any locality of the Hamiltonian. Applicationsin quantum time crystals are briefly discussed.

Absence of temporal order in states with spatial correlation decay