High Quality crystals are Exclusive so they can't be traded or sold. But you can have more than one (they stack to 12). The only difference between these and the normal crystals you can get from mobs and the Auction House is that the High Quality Crystal will add your name at the bottom of the little display window of the item you make. They don't help you succeed, or give a higher chance for a high quality synth.
There are catches - it won't put your name on just anything. You can only sign a "final product' item. You can't sign stackable items, and you can't sign a material. Popular items to put your signature on are armor, weapons, furniture and the cute Lv.1 recipe Love Chocolate.
If you sell your signed item at the Auction House then your signature will disappear. So if you want to keep your name on it, sell it in your bazaar or trade it to someone. The signature on an armor piece will also disappear if the armor is stored with an armor storage NPC.
Cost in GPs