Part of the interest in topological phases of matter stems from their potentially revolutionary technological applications. One of the most exciting possibilities would be the ability to store and manipulate quantum information in topological degrees of freedom. This information would be intrinsically protected from some local error processes and there are now significant efforts being made to harness this property in scalable quantum devices. In this context, we numerically investigate the dynamical evolution of a topological memory that consists of two p-wave superconducting wires separated by a non-topological junction. We consider a closed system and do not consider general types of errors that may arise from the interaction of the system with an external environment. We therefore consider the best case scenario apt to quantum computation and expand upon existing knowledge on the subject. We find that non-adiabatic effects associated with processing and storing quantum information, might be relevant as a source of errors in spite of the topological protection of the quantum qubit. In particular we focus on the primary source of errors (i.e., qubit-loss) and on secondary errors (bit and phase- ip) and provide physical reasoning behind how these types of errors may take place. Further we consider model for more exotic quantum wires, the so-called ZN parafermionic clock models that are also expected, in varying degrees, to possess topological protection. We analyze in detail when this protection might take place and provide an iterative construction of the associated zero-energy parafermionic modes. The method, outlined in this thesis, summarizes and generalizes existing methods for constructing zero modes in spin chains and enlightens common surprising features of these types of constructions. This opens up the way for future research on the subject.