There are algorithms that can have an exponential speedup on a quantum computer, there are plenty that have a quadratic speedup, for example, searching an unsorted/unindexed database takes O(sqrt(n)) instead of O(n) (with high probability of success). Which means that any optimization problem (with a search tree) can have this quadratic speedup too.

Quantum cryptography has some pretty lovely nature tricks that enable no chance of spying/hacking, measuring the state plays so weirdly with the quantum system.