  We prove that the solution map, associated to the quintic fourth order
nonlinear Schr\"odinger equation, exhibits the norm inflation phenomenon at
every point in the Sobolev spaces of super-critical regularity. Indeed, we
prove this result separately in the cases of negative and of positive
regularity. In the negative regularity case, we prove the result for both the
defocusing and focusing equations: in the one dimensional case, the associated
solution map exhibits norm inflation in Sobolev spaces of super-critical
regularity (including the critical index); in the higher dimensional case, the
solution map exhibits the same phenomenon in spaces of negative regularity.
Meanwhile, in the case of positive regularity, we prove the result for the
defocusing equation in dimensions $d=3,4,5$. Our proofs are based on the
"high-to-low" and "low-to-high" frequency cascades respectively.
