Quantum cohomology and birational geometry of Verra fourfolds

We compute the small quantum cohomology ring of a Verra fourfold. Using the theory of atoms recently developped by Katzarkov–Kontsevich–Pantev–Yu, and building on recent papers of the authors, we deduce that a Verra fourfold is never birational to a very general cubic fourfold, nor to a very general Gushel–Mukai fourfold, whereas it was previously known that a general Verra fourfold is birational to a general nodal Gushel–Mukai fourfold. More precisely, we show that for every smooth cubic fourfold or smooth Gushel–Mukai fourfold that is birational to some Verra fourfold, the primitive cohomology is isomorphic, as a rational Hodge structure, to the middle cohomology of some projective K3 surface.