We consider Clifford quantum cellular automata (CQCAs) and their time-evolution. CQCAs are an especially simple type of quantum cellular automata, yet they show complex asymptotics and can even be a basic ingredient for universal quantum computation. In this work we study the time evolution of different classes of CQCAs. We distinguish between periodic CQCAs, fractal CQCAs, and CQCAs with gliders. We then identify invariant states and study convergence properties of classes of states, such as quasifree and stabilizer states. Finally, we consider the generation of entanglement analytically and numerically for stabilizer and quasifree states.

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