  We present a coarse perspective on relations of the
$SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the
Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the
asymptotic expansions of the curve operators in the skein theoretic version of
the $SU(2)$-TQFT, as developed by Blanchet, Habegger, Masbaum and Vogel, we
define the quantum intersection number between pants decompositions of a closed
surface. We show that the quantum intersection number admits two sided bounds
in terms of the geometric intersection number and we use it to obtain a metric
on the pants graph of surfaces. Using work of Brock we show that the pants
graph equipped with this metric is quasi-isometric to the Teichm\"uller space
with the Weil-Petersson metric and that the translation length of our metric
provides two sided linear bounds on the volume of hyperbolic fibered manifolds.
  We also obtain a characterization of pseudo-Anosov mapping classes in terms
of asymptotics of the quantum intersection number under iteration in the
mapping class group and relate these asymptotics with stretch factors.
