Ancient relics that keep the Incarnates imprisoned in time via Malphas’s spell. It seems that there are 7 hearts, one per incarnate, each granted to one of the Dawnborn’s loyal mortal kingdoms as signs of trust to be defended against the darkness. They may also have additional magical powers and access. Supposedly, if all the hearts are used to break Malphas’s barrier the world will end.
When Lysandre went mad, he believed that the Holy Order was coveting the orb and that enemies were abound attempting to steal it. After having suffered the loss of one by Lysandre, the others had increased security.
If the barrier broke it would be the end of the world.