The registration process for Appendix QH's can get a little confusing. I had to look it up again on AQHA's website to make sure I had all my facts straight.
So, when a registered QH is bred to a Jockey club TB the resulting foal is registered under the AQHA's appendix registry. The appendix horse is still eligible to compete in all AQHA events. The only time you come across restrictions is if you plan on breeding the animal. Appendix horses can only be bred to non-appendix Quarter horses in order for the foal to be registered. I'm pretty sure that foal will also be registered as an appendix.
However! AQHA does allow exceptions to this rule. If the horse earns a ROM in the open division at shows or achieves an 80 or better Speed Index on a race track, the horse is eligible to advance into the regular AQHA registry. This means all their offspring are registered under the regular AQHA registry as well, and can then be bred back to a TB or appendix QH for an appendix foal.
As a result of all this confusion, it is possible to end up with a registered appendix horse that is more thoroughbred than quarter horse.