  We consider families of quasisymmetric functions with the property that if a
symmetric function $f$ is a positive sum of functions in one of these families,
then f is necessarily a positive sum of Schur functions. Furthermore, in each
of the families studied, we give a combinatorial description of the Schur
coefficients of $f$. We organize six such families into a poset, where
functions in higher families in the poset are always positive integer sums of
functions in each of the lower families. This poset includes the Schur
functions, the quasisymmetric Schur functions, the fundamental quasisymmetric
generating functions of shifted dual equivalence classes, as well as three new
families of functions --- one of which is conjectured to be a basis of the
vector space of quasisymmetric functions. Each of the six families is realized
as the fundamental quasisymmetric generating functions over the classes of some
refinement of dual Knuth equivalence. Thus, we also produce a poset of
refinements of dual Knuth equivalence. In doing so, we define quasi-dual
equivalence to provide classes that generate quasisymmetric Schur functions.
