In this talk, $H(\mathrm{curl}^2)$-conforming elements are constructed explicitly in a hierarchical pattern. We propose our $H(\mathrm{curl}^2)$-conforming spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Notably, the constructed elements can be categorized into the kernel space and non-kernel space of the curl operator. The degrees of polynomials of the kernel space only affect the convergence rate of the $L^2(\Omega)$-norm of uh but not the semi-norm of $H(\mathrm{curl};\Omega)$ and $H(\mathrm{curl}^2;\Omega)$. This allows us to obtain eigenvalue approximations from the upper or lower side by choosing different degrees of polynomials for the kernel space and non-kernel space of the curl operator. Numerical results show the effectiveness and efficiency of our method.