Summary: Quantum Hall universality classes can be classified by \(W_{1+ \infty}\) symmetry. We show that this symmetry also governs the dynamics of quantum edge excitations. The Hamiltonian of interacting electrons in the fully-filled first Landau level is expressed in terms of \(W_{1+ \infty}\) generators. The spectra for both the Coulomb and generic short-range interactions are thus found algebraically. We prove the one-dimensional bosonization of edge excitations in the limit of large number of particles. Moreover, the subleading corrections are given by the higher-spin \(W_{1+ \infty}\) generators, which measure the radial fluctuations of the electron density. The resulting spectrum for the Coulomb interaction contains a logarithmic enhancement, in agreement with experimental observations. Generic short-range interactions yield a subleading contribution to the spectrum, which can be expressed in terms of the classical capillary frequencies. These results are also extended to the Laughlin fractional fillings \(\nu= 1/3\), \(1/5,\dots\) by using symmetry arguments.