  In this paper we study real hypersurfaces in the complex quadric space $Q^m$
whose structure Jacobi operator commutes with their structure tensor field. We
show that the Reeb curvature $\alpha$ of such hypersurfaces is constant and if
$\alpha$ is non-zero then the hypersurface is a tube around a totally geodesic
submanifold $\mathbb{C} P^k \subset Q^m$, where $m=2k$. We also consider Reeb
flat hypersurfaces, namely, when the Reeb curvature is zero. We show that the
tube around $\mathbb{C} P^k \subset Q^m$ ($m=2k$), with radius $\frac{\pi}{4}$
is the only Reeb flat Hopf hypersurface with commuting Ricci tensor and also
the only one with commuting shape operator. Finally, we prove that there does
not exist any Reeb flat Hopf hypersurfaces with non-parallel Killing Ricci
tensor or with Killing shape operator.
