The two dimensional electron gas in the regime of the Fractional Quantum Hall Effect is one of the hallmarks of condensed matter physics. One of its main characteristic features is that quasiparticle excitations of this system exhibit both fractional charge and fractional "anyonic" exchange statistics. Experiments involving resonant quasiparticle tunnelling through Quantum Antidots have demonstrated the potential to manipulate individual quasiparticles. It is also known that the anyonic exchange statistics of the quasiparticles can be exploited for use in Quantum Information. The basic building block for this type of Quantum Information processing is the FQHE qubit which is formed from two tunnel coupled quantum antidots. In the first part of this dissertation, a model describing the coherent tunnelling of quasiparticles of quantum hall liquids in a system consisting of multiple antidots will be discussed. The main result is that the anyonic exchange statistics of these quasiparticles is manifested directly in the DC tunnel conductance of these systems even in the absence of quasiparticle exchange. Most notably, it will be shown that in tunnelling through a line of three antidots, the statistics should be exhibited as a non-vanishing resonant peak of the tunnel conductance. The second half will be dedicated in part to exploring the potential use of FQHE qubits in applications involving Quantum Information. To begin with, the Quantum Antidot Electrometer will be discussed as a detector for quantum measurements of FQHE qubits. Next, the non-trivial aspects of wave function reduction will be examined as well as the coherent synchronization of oscillations in a continuously measured double qubit system. The dissertation concludes with an examination of a different paradigm in Quantum Information processing namely that of adiabatic quantum computation (AQC). Due to the ground state evolution of AQC it is expected that this scheme provides a measure of protection against environmental decoherence. The stability of this scheme of quantum computation is assessed with respect to decoherence induced by low frequency noise which is of particular relevance to solid state implementations of AQC.