Here’s my thoughts on the weird crystals in Sun and Moon:
Everyone’s so wrapped up in thinking that they are the Burst Crystals from an obscure Pokemon manga from gen 5, but I think differently about this.
TLDR: Solgaleo and Lunala energize each other. The yellow/blue crystal emits Solgaleo/Lunala’s energy, and is used to summon Lunala/Solgaleo. The crystals are also used to fuel Mega Evolution (if it still exists), as well as provide a new mechanic involving Pokemon moves, which Prof. Kukui researches.