  We show that the identity component of the group of diffeomorphisms of a
closed oriented surface of positive genus admits many unbounded
quasi-morphisms. As a corollary, we also deduce that this group is not
uniformly perfect and its fragmentation norm is unbounded, answering a question
of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph
on which these groups act, which is the analog of the curve graph for the
mapping class group.
