  In this paper, we propose an $H(\text{curl}^2)$-conforming quadrilateral
spectral element method to solve quad-curl problems. Starting with generalized
Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for
vector fields over rectangles. $H(\text{curl}^2)$-conforming elements over
arbitrary convex quadrilaterals are then constructed explicitly in a
hierarchical pattern using the contravariant transform together with the
bilinear mapping from the reference square onto each quadrilateral. It is
astonishing that both the simplest rectangular and quadrilateral spectral
elements possess only 8 degrees of freedom on each physical element. In the
sequel, we propose our $H(\text{curl}^2)$-conforming quadrilateral spectral
element approximation based on the mixed weak formulation to solve the
quad-curl equation and its eigenvalue problem. Numerical results show the
effectiveness and efficiency of our method.
