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Here we define the concept of Qregularity for coherent sheaves on quadrics.In this setting we prove analogs of some classical properties. We compare theQregularity of coherent sheaves on $\Q_n\subset \mathbb P^{n+1}$ with theCastelnuovo-Mumford regularity of their extension by zero in $\mathbb P^{n+1}$.We also classify the coherent sheaves with Qregularity $-\infty$. We use ournotion of Qregularity in order to prove an extension of Evans-Griffithscriterion to vector bundles on Quadrics. In particular we get a new and simpleproof of the Kn\"{o}rrer's characterization of ACM bundles.

Qregularity and an Extension of Evans-Griffiths Criterion to Vector Bundles on Quadrics