  We show that the Nielsen-Thurston classification of mapping classes of the
sphere with four marked points is determined by the quantum
SU(n)-representations, for any fixed integer $n \geq 2$. In the Pseudo-Anosov
case we also show that the stretching factor is a limit of eigenvalues of
(non-unitary) SU(2)-TQFT representation matrices. It follows that at big enough
levels, Pseudo-Anosov mapping classes are represented by matrices of infinite
order.
