  We study the existence of faithful tracial states on C*-algebras as well as
the stronger proerty that all quotients admit faithful tracial states. We
provide a sufficient and necessary condition for when C*-algebras admit
faithful tracial states in terms of the Cuntz semigroup and use this to give an
equivalent formulation for all quotients to admit faithful tracial states. We
relate this to the notion of strong quasidiagonality, and show that any
amenable discrete group with faithful tracial states on all quotients of the
corresponding group-C*-algebra is strongly quasidiagonal under the condition
that all quotients satisfy the Universal Coefficient Theorem.
