  We prove and organize some results on the normal forms of Hermitian operators
composed with the Veronese map. We apply this general framework to prove two
specific theorems in CR geometry. First, extending a theorem of Faran, we
classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any
hyperquadric in $\C^3$, resulting in a finite list of equivalence classes.
Second, we prove that all degree-two CR maps of spheres in all dimensions are
spherically equivalent to a monomial map, thus obtaining an elegant
classification of all degree-two CR sphere maps.
