If we extend the definition of "computing" slightly to include "coding and cryptography", then I find superdense coding quite exciting [0]. In a hypothetical quantum internet we could encode and transmit two classical bits in every qubit, which effectively doubles your bandwidth. There's also quantum key distribution [1]. In principle, quantum theory allows Alice and Bob to establish a shared private key which cannot be broken without breaking the laws of physics. You can chat privately with people secure in the knowledge that not even the NSA can possibly be listening.