  Let $G$ be a connected, simply connected, simple, complex, linear algebraic
group. Let $P$ be an arbitrary parabolic subgroup of $G$. Let $X=G/P$ be the
$G$-homogeneous projective space attached to this situation. We consider the
(small) quantum cohomology ring $(QH^*(X),\star)$ attached to $X$. We prove
that there exists a unique degree $d$ which is minimal with the property that
$q^d$ occurs with non-zero coefficient in the quantum product of two point
classes. We denote this minimal degree in $\mathrm{pt}\star\mathrm{pt}$ by
$d_X$. We give an explicit formula to compute $d_X$ in terms of the cascade of
orthogonal roots. We construct an explicit curve of degree $d_X$ passing
through two general points in $X$. Moreover, we prove that $d_X$ is the unique
maximal element of the set of all minimal degrees in some quantum product of
two Schubert classes.
