The Higgs boson, as I understand it, is THE elementary particle from which all other particles are formed. They were present during the beginning of the big bang, and it is THEIR existence, their unique compositions and iterations, that form all matter, antimatter, and what have you in the universe.
A quantum, as I understand it, is the smallest amount of energy required to make fundamental particles interact. According to the standard model, quanta are responsible for all motion and movement in the universe. We KNOW that all matter is in motion, all the time - just at varying magnitudes. The only time that the universe wasn't in motion was before the big bang; when everything was as its zero point.
So what caused the big bang? Theoretically, a single quantum could have done it, like a domino effect. What set that quantum (or those quanta) in motion? This is pure speculation, mind you, but I think that if we can figure out what CAUSES quantum mechanics, then we've solved the question - how did we get here? What set the universe in motion?
Physics is SO much fun to think about =)