  We analyze the phase diagram of the zeroth Landau level of bilayer graphene,
taking into account the realistic effects of screening of the Coulomb
interaction and the strong mixing between two degenerate sublevels. We identify
robust quantum Hall states at filling factors \nu=-1, -4/3, -5/3, -8/5, -1/2,
and discuss the nature of their ground states, collective excitations, and
relation to the more familiar states in GaAs using a tractable model. In
particular, we present evidence that the \nu=-1/2 state, which was recently
reported experimentally, is non-Abelian, and described by either the Moore-Read
Pfaffian wave function or its particle-hole conjugate, while ruling out other
candidates such as the 331 state.
