  The $n$-dimensional complex hyperquadric is a compact complex algebraic
hypersurface defined by the quadratic equation in the $(n+1)$-dimensional
complex projective space, which is isometric to the real Grassmann manifold of
oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In
this paper we study geometry of compact Lagrangian submanifolds in complex
hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces
in spheres. From this viewpoint we provide a classification theorem of compact
homogeneous Lagrangian submanifolds in complex hyperquadrics by using the
moment map technique. Moreover we determine the Hamiltonian stability of
compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which
are obtained as Gauss images of isoparametric hypersurfaces in spheres with
$g(=1,2,3)$ distinct principal curvatures.
