Optical binding refers to the light-induced interaction between two or more objects illuminated by laser fields. The high tunability of the strength, sign, and reciprocity of this interaction renders it highly attractive for controlling nanoscale mechanical motion. Here, we discuss the quantum theory of optical binding and identify unique signatures of this interaction in the quantum regime. We show that these signatures are observable in near-future experiments with levitated nanoparticles. In addition, we prove the impossibility of entanglement induced by far-field optical binding in free space and identify strategies to circumvent this no-go theorem.