In the thesis, models of quantum systems characterized by discrete time dynamics are presented and applied to different problems. More specifically, aspects of excitation transport, quantum state transfer and thermalization are investigated mainly within the framework of quantum cellular automata (QCA), which are discrete time evolution quantum systems defined on a regular lattice of, in this case, qubits. The peculiarities of QCA, together with the restriction to the first excitation sector of the Hilbert space, make it possible to define a transport dynamics on a one dimensional lattice that encompasses all classical Markov chains, as well as maps where quantum coherence between sites can build up over time. This allows for the possibility to make a direct comparison between a fully classical dynamics and a dynamics affected by quantum effects, which is of relevance both in the field of quantum biology, especially regarding the transport of electronic excitations in photosynthetic systems, and in quantum computing when dealing with the transport of an unknown quantum state. In a quantum thermodynamics context, instead, QCA can serve the purpose to validate arguments of tipicality that have been brought forward in recent years in order to demonstrate the thermalization of closed quantum systems (i.e. the tendency of any small subsystems to evolve towards the maximally mixed state). Numerical results concerning this issue are presented. Finally, a modelization of scattering-like processes, where the scattering consists of random unitary interactions between an environment and part of a quantum bipartite system is discussed. Relevant properties of the resulting state such as its mean purity and the correlations between the interacting and the “sheltered” part of the system can be analitically evaluated, and their fluctuations bounded
