Mammalian cells have two fundamentally different states – proliferative and quiescent – but our understanding of how and why cells switch between these states is limited. We previously showed that actively proliferating populations contain a subpopulation that enters quiescence (G0) in an apparently stochastic manner. Using single-cell time-lapse imaging of CDK2 activity and DNA damage, we now show that endogenous replication stress in the previous (mother) cell cycle prompts p21-dependent entry of daughter cells into quiescence immediately after mitosis. Furthermore, the amount of time daughter cells spend in quiescence is correlated with the extent of inherited damage. Our study thus links replication errors in one cell cycle to the fate of daughter cells in the subsequent cell cycle. More broadly, this work reveals that entry into quiescence is not purely stochastic but has a strong deterministic component arising from a memory of events that occurred in the previous generation(s).