  We discuss a surprising relationship between the partially ordered set of
Newton points associated to an affine Schubert cell and the quantum cohomology
of the complex flag variety. The main theorem provides a combinatorial formula
for the unique maximum element in this poset in terms of paths in the quantum
Bruhat graph, whose vertices are indexed by elements in the finite Weyl group.
Key to establishing this connection is the fact that paths in the quantum
Bruhat graph encode saturated chains in the strong Bruhat order on the affine
Weyl group. This correspondence is also fundamental in the work of Lam and
Shimozono establishing Peterson's isomorphism between the quantum cohomology of
the finite flag variety and the homology of the affine Grassmannian. One
important geometric application of the present work is an inequality which
provides a necessary condition for non-emptiness of certain affine
Deligne-Lusztig varieties in the affine flag variety.
