Let A and B be two connected graded commutative k–algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi–shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k–vector spaces equipped with a Frobenius (pth–power) map.For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split as a morphism of graded k–vector spaces equipped with a Verschiebung map.Building on work on non–commutative Witt vectors by Goerss, Lannes, and Morel, we classify such free, ‘split’ Hopf algebras.A topological consequence is that, if X is a based path connected space, then the Hopf algebra H⁎(ΩΣX;k) is determined by the stable homotopy type of X.We also discuss the much easier analogous characteristic 0 results, and give a characterization of when our quasi–shuffle algebras are polynomial, generalizing the so–called Ditters conjecture.