Transition from Abelian to non-Abelian quantum liquids in the second Landau level

The search for non-Abelian quantum Hall states in the second Landau level is narrowed to the range of filling factors 7/3<\nu_e<8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at \nu_e=5/2 (Moore--Read) and 12/5 (Read--Rezayi). Outside of it, the four-flux noninteracting composite fermion model is validated. The borderline \nu_e=7/3 state is adiabatically connected to the Laughlin liquid, but its short-range correlations and charge excitations are different.