The detection of non-Abelian exchange statistics is an open challenge which holds important promises for the advent of topological quantum computation. A recent work proposes to rely on the edges to reveal the braiding statistics of nonAbelian anyons in the bulk, in an entirely deterministic dynamical process. A time-dependent gap in a Josephson junction couples two co-propagating Majorana fermions, and as the gap closes, a pair of edge-vortices is injected into the edges. Because these defects have the same non-Abelian statistics, they are braided with vortices in the bulk. Conveniently, the fusion of the edge-vortices results in a quantized unit of charge at the exit. However, this process is so far only predicted in the adiabatic limit. In this work, this assumption is relaxed by means of a full manybody evolution of the superconducting ground state in the Bogoliubov-de-Gennes formalism. Beyond revealing the collective nature of the edge-vortex excitation, we demonstrate that the quantization of charge still holds if the system does not return to the ground state. Furthermore, the effect of path length difference between the edge-vortices confirms the theoretical predictions done in another work on the subject. At fast injections, we reveal weak oscillations in current contributed by the bound states in the junction which average to zero and are removed in the short junction limit. This work is concluded with a preliminary evaluation of the manybody parity operator, which indicates that the edge-vortex may encode the parity of the bulk vortices. This opens the possibility for sequential qubit manipulations on the edge-vortex.

The edge states in finite quantum Hall graphene have previously been shown to be valley polarised for zigzag and armchair edges. Assuming that the valley isospin is also conserved at a smooth normal-superconducting (NS) interface, theoretical research has previously predicted that plateaus in the longitudinal conductance are expected to occur in the lowest Landau level of the incoming edge modes, which depends on the angle difference between the isospins entering and leaving the superconductor. In this thesis, this prediction is verified with a tight-binding simulation of six different NS junctions: for both zigzag and armchair edge nanoribbons, the superconductor can cover a single edge, two adjacent edges or two opposite edges. The theoretical prediction could be confirmed successfully, suggesting that the edge states are valley polarised along a smooth NS interface. Some deviations from the theory could be observed for the armchair edge ribbon with opposite edges when the width of the ribbon is not a multiple of three hexagons. Two consecutive widths show a complementary behaviour in the conductance such that their average corresponds to the predicted value with a remarkable robustness. The reason for this complementarity was briefly conjectured by using the special Andreev reflection in graphene and the coupling between sublattice and valley degree of freedom for zigzag edges. The parameter regimes allowing for the existence of conductance plateaus were established, confirming that the plateaus emerge for a system size much larger than the magnetic length and the superconducting coherence length, and that a smooth chemical potential, magnetic field strength and superconducting order parameter are necessary at the NS interface. The robustness of the NS edge states was furthermore investigated with three methods: a Fermi energy mismatch between the bulk and the superconductor, and random normally distributed variation in the onsite electrostatic potential and a random potential landscape. All results could confirm that intervalley scattering is the reason for deviations from the plateaus predicted by the theory.