  We prove that under a symmetry assumption all cocycles on Hopf *-algebras
arise from generating functionals. This extends earlier results of R.Vergnioux
and D.Kyed and has two quantum group applications: all quantum L\'evy processes
with symmetric generating functionals decompose into a maximal Gaussian and
purely non-Gaussian part and the Haagerup property for discrete quantum groups
is characterized by the existence of an arbitrary proper cocycle.
