  We prove a bound for the geodesic diameter of a subset of the unit ball in
$\mathbb{R}^n$ described by a fixed number of quadratic equations and
inequalities, which is polynomial in $n$, whereas the known bound for general
degree is exponential in $n$. Our proof uses methods borrowed from D'Acunto and
Kurdyka (to deal with the geodesic diameter) and from Barvinok (to take
advantage of the quadratic nature).
