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This review informs that the impossibility of genuine quantum time crystalshas been known in the C*-algebraic quantum statistical mechanics since 1970s.The KMS condition implies that spontaneous breakdown of time-translationsymmetry is impossible in general quantum systems. In particular, anynon-trivial order, such as periodic, quasi-periodic, and chaotic order cannotexist in the time direction of equilibrium states, meaning that genuine quantumtime crystals are excluded from the outset.

Nonexistence of spontaneous symmetry breakdown of time-translation symmetry on general quantum systems: Any macroscopic order parameter moves not!