  We prove that on a nonsingular quadric hypersurface Q in $P^4_k$ even CI
liaison classes of ACM curves are in bijective correspondence with the stable
equivalence classes (up to shift in degree) of maximal Cohen-Macaulay graded
modules over the coordinate ring R of Q, which in turn, are in bijective
correspondence with stable equivalence classes (up to shift in degree) of ACM
sheaves on Q . In the situation of a nonsingular quadric hypersurface Q in
$P^4_k$ work of Knorrer shows that there is a unique nonfree indecomposable MCM
module over R. We also describe the ACM sheaf corresponding to this MCM module
and give its cohomology table and its Hilbert polynomial.
