  This paper deals with the notion of quadratic differential in spherical CR
geometry (or more generally on strictly pseudoconvex CR manifolds). We get to
this notion by studying a splitting of Rumin complex and discuss its first
features such as trajectories and length. We also define several differential
operators on quadratic differentials that lead to analogue of half-translation
structures on spherical CR manifolds. Finally, we work on known examples of
quasiconformal maps in the Heisenberg group with extremal properties and
explicit how quadratic differentials are involved in those.
  In addition, on our way to quadratic differentials, we define a differential
complex on strictly pseudoconvex CR manifolds with a finite dimensional
cohomology space. It leads to a new CR invariant that we compute for compact
manifolds endowed with a CR action of the circle.
