When a two-dimensional electron gas is cooled to cryogenic temperatures and subjected to a strong perpendicular magnetic field, the Hall resistance becomes quantized. When this quantization occurs at fractional values, it signals the fractional quantum Hall effect (FQHE), a strongly correlated phase of matter with emergent topological order. A striking consequence is the appearance of fractionally charged particles, called anyons, now observed experimentally, whose exchange can yield fractional statistics. In special FQHE states, these anyons are non-Abelian, providing a route to realizing fault-tolerant topological quantum computation. While such properties are long predicted by low-energy Chern–Simons and conformal field theory descriptions, obtaining microscopic access, directly in terms of electronic coordinates and explicit many-body wave functions, remains challenging beyond a few canonical examples. In this thesis, I develop many-body wave-function frameworks to study FQH phases, ground states, and excitations at the microscopic level. This includes analyzing both collective (neutral) excitations, Abelian and non-Abelian anyons, and connecting their properties to the underlying topological field theory. In this talk, I will give a brief overview of the above topics and then highlight a work on the construction of quasihole bases for a broad family of non-Abelian FQH states using parton wave functions. This reproduces the fusion-space dimensionality expected from their underlying conformal field theory, consistent with level-rank duality across the parton family. As an application, we numerically compute braiding matrices for representative parton states for large systems, providing a general framework for diagnosing non-Abelian characteristics in candidate FQH states.

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