In this thesis, we theoretically study low energy excitations in two types of quantum condensates, Bose-Einstein condensates and Quantum Hall condensates. First, we investigate the effect of an anisotropic trap on the instability of the polar (m_F = 0) phase of a spin-1 Bose-Einstein condensate. By rigorously considering the spatial quantization, we show that the growth of the nascent ferromagnetic phase at short times becomes anisotropic with stronger oscillations in the magnetization correlation function along the unconfined direction. Turning to quantum Hall condensates, we analyze edge excitations in the ν = 1 plateau and demonstrate that two experimentally observed features of edge modes, rapid head decay and long-lived charged excitations, cannot be consistently explained using a model where energy is dissipated from a hard edge via either phonon interactions or bulk AC conductivity. We suggest a soft edge model where heat can decay through intra-edge interactions to explain the discrepancy and calculate the scaling properties of this model. We then demonstrate numerically that non-Abelian quasihole excitations of the ν = 5/2 fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the quasihole spacing is increased, the two orthogonal wavefunctions describing a system with four quasiholes become exponentially degenerate with decay length ξ_E≈2.3ℓ₀. Additionally we determine which fusion channel is lower in energy when two quasiholes are brought close together. Finally, we consider a hypothetical topological quantum computer composed of either Ising or Fibonacci anyons. For each case, we calculate the time and number of qubits necessary to execute the most computationally expensive step of Shor's quantum factoring algorithm, modular exponentiation. With reasonable restrictions on the physical parameters we find that factoring a 128 bit number requires approximately 10³ Fibonacci anyons versus at least 3 x 10⁹ Ising anyons.