When confined to a finite, two-dimensional area and exposed to a strong magnetic field, electrons exhibit a complicated, highly correlated quantum behavior known as the quantum Hall effect. This dissertation consists of finite size numerical investigations of this effect. One line of study develops treatment of the fractional quantum Hall effect using the hyperspherical method, in conjunction with applications to the few-body quantum Hall systems, e.g., highly-controlled atomic systems. Another line of research fully utilizes the developed numerical techniques to study on the platform of finite size fractional quantum Hall states the bulk-edge correspondence principle, which is universal for phases in topological orders. It has been demonstrated that the eigenstates associated with the entanglement spectrum reveal more information about the ground state than the spectrum alone.