You've already got your PhD, right?
The qualifications are to be funny and to have some major, major self-esteem issues. It really doesn't matter how many classes you've taken, what books you've read on the art of stand-up, or even who you know. You just have to get up on that stage, stand behind that mic, and make people laugh until they're purple. If you can do that on a regular basis, you may actually be able to move up in the world of comedy.
Otherwise, you will probably have to settle for being a joke.