A kind of quantum metasurface made of an atom array is proposed, providing the possibility to control both spatiotemporal and quantum properties of transmitted and reflected light. Metasurfaces mould the flow of classical light waves by engineering subwavelength patterns from dielectric or metallic thin films. We introduce and analyse a method in which quantum operator-valued reflectivity can be used to control both the spatiotemporal and quantum properties of transmitted and reflected light. Such quantum metasurfaces are realized by entangling the macroscopic response of atomically thin atom arrays to light. We show that such a system allows for parallel quantum operations between atoms and photons as well as for the generation of highly entangled photonic states such as photonic Greenberger-Horne-Zeilinger and three-dimensional cluster states suitable for quantum information processing. We analyse the influence of imperfections as well as specific implementations based on atom arrays excited into Rydberg states.