  In this paper we study the prescribed fractional $Q$-curvatures problem of
order $2 \sigma$ on the $n$-dimensional standard sphere $(\mathbb{S}^{n},
g_0)$, where $n\geq3$, $\sigma\in(0,\frac{n-2}{2})$. By combining critical
points at infinity approach with Morse theory we obtain new existence results
under suitable pinching conditions.
