The gems being associated with brains, power, and heart was the first thing I noticed in the book. (obviously implying the three girls).

I find it really interesting how the way the box is powered seems to be through the magical embodiments of certain character traits. I assume, since each trait also applies to a respective race within Amphibia, that maybe it was built that way so, for a gateway to another dimension to be opened, all three main races of Amphibia had to be unified and willing to do so.

This would probably be to prevent 1 race from getting ahold of it and activating it by themselves, and by default make them the dominant society within Amphibia. If 1 race could activate it by themselves, then the most likely outcome would be a never-ending civil war to obtain the box. Or maybe there was a war over it, and the creation of each respective temple/gem was a way to induce equality and regain peace within Amphibia.