The divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under infinitesimal perturbations, and characterize Pauli dynamical semigroups exhibiting such a property. We construct collision models with factorized environment particles, which realize additivity and multiplicativity of generators of CP divisible maps. A mixture of dynamical maps is obtained with the help of correlated environment. The mixture of ultimate CP divisible processes is shown to result in a class of eternal CP indivisible evolutions. We explicitly find collision models leading to weakly and essentially non-Markovian Pauli dynamical maps.

We show that it is possible to use a quantum walk to find a path from one marked vertex to another. In the specific case of M stars connected in a chain, one can find the path from the first star to the last one in O(M√N) steps, where N is the number of spokes of each star. First we provide an analytical result showing that by starting in a phase-modulated highly superposed initial state we can find the path in O(M√N logM) steps. Next, we improve this efficiency by showing that the recovery of the path can also be performed by a series of successive searches when we start at the last known position and search for the next connection in O(√N) steps leading to the overall efficiency of O(M√N). For this result we use the analytical solution that can be obtained for a ring of stars of double the length of the chain.