# What are the differences and similarities between Quantum synchronization and time-crystalisation?

I am learning about Quantum synchronization and read [1,2] on Sunday, which are about many-body bosonic systems. I understand that what happens when the system synchronizes is that all the particles in the system start oscillating in phase. In practice this happens when the destruction and creation operators $$a$$ and $$a^\dagger$$ acquire a finite expectation value, and the system spontaneously breaks $$U(1)$$ symmetry. This happens in a driven-dissipative set-up.

On the other hand systems that are periodically driven can become time crystals and spontaneously break the discrete time-translation symmetry. I expect from the literature that this can not happen if time-translation symmetry is not already broken to a discrete one because nobody talks about it, but am not sure. Does the equilibrium argument apply to a nonequilibrium (driven-dissipative) set-up?

Both phenomena (quantum synchronization and time crystals) share the same idea of an emergent periodic time dependence, but the terminology (and literature) seem very different to me. I can see one difference between them in the fact that time-translation is explicitly broken (to a discrete symmetry) for time crystals but not for quantum synchronization, but what are the similarities and are there other differences between the two phenomena?