Quantum time crystal has been an intriguing many-body “time” state that has received much attention and debate since its early prediction. In this talk, first, I will construct a class of concrete “clean” Floquet models to answer the open question on the role of disorder and many-body localization. Second, by observing the equivalent role of the space and imaginary time in the path integral formalism, I will present the finding that hard-core bosons coupled to a thermal bath may exhibit the order of “imaginary spacetime crystal”.