In this paper, we pursue our analysis of the W 1+∞ symmetry of the low-energy edge excitations of incompressible quantum Hall fluids. These excitations are described by (1 + 1)-dimensional effective field theories, which are built by representations of the W 1+∞ algebra. Generic W 1+∞ theories predict many more fluids than the few, stable ones found in experiments. Here we identify a particular class of W 1+∞ theories, the minimal models, which are made of degenerate representations only - a familiar construction in conformal field theory. The W 1+∞ minimal models exist for specific values of the fractional Hall conductivity, which nicely fit the experimental data and match the results of the Jain hierarchy of quantum Hall fluids. We thus obtain a new hierarchical construction, which is based uniquely on the concept of quantum incompressible fluid and is independent of Jain's approach and hypotheses. Furthermore, a surprising non-abelian structure is found in the W 1+∞ minimal models: they possess neutral quark-like excitations with SU( m) quantum numbers and non-abelian fractional statistics. The physical electron is made of anyon and quark excitations. We discuss some properties of these neutral excitations which could be seen in experiments and numerical simulations.