  In this paper we study the existence of gradings on finite dimensional
associative algebras. We prove that a connected algebra $A$ does not have a
non-trivial grading if and only if $A$ is basic, its quiver has one vertex, and
its group of outer automorphisms is unipotent. We apply this result to prove
that up to graded Morita equivalence there do not exist non-trivial gradings on
the blocks of group algebras with quaternion defect groups and one isomorphism
class of simple modules.
