  Noncommutative hypersurfaces, in particular, noncommutative quadric
hypersurfaces are major objects of study in noncommutative algebraic geometry.
In the commutative case, Kn\"orrer's periodicity theorem is a powerful tool to
study Cohen-Macaulay representation theory since it reduces the number of
variables in computing the stable category $\underline{\operatorname{CM}}(A)$
of maximal Cohen-Macaulay modules over a hypersurface $A$. In this paper, we
prove a noncommutative graded version of Kn\"orrer's periodicity theorem.
Moreover, we prove another way to reduce the number of variables in computing
the stable category ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ of graded
maximal Cohen-Macaulay modules if $A$ is a noncommutative quadric hypersurface.
Under high rank property defined in this paper, we also show that computing
${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a noncommutative smooth
quadric hypersurface $A$ in up to six variables can be reduced to one or two
variables cases. In addition, we give a complete classification of
${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a smooth quadric
hypersurface $A$ in a skew $\mathbb P^{n-1}$, where $n \leq 6$, without high
rank property using graphical methods.
