No. This statement is false.
You are confusing two different things. The movement of quantum states from one energy eigenstate to another is not what I'm talking about.
Matter and energy literally appear from fluctuations of the vacuum state, in which there is NOTHING in conventional terms. [And I'm not surprised you don't know this, since you clearly know nothing about quantum mechanics.] Get started here, since you've clearly never heard of this: http://en.wikipedia.org/wiki/Quantum_fluctuation
It is my understanding that gravitational singularity will over time evaporate away (Hawken's Radiation) not explode.
Your understanding is of black holes, which have collapsed in ordinary spacetime. But the singularity at the beginning of spacetime was not a gravitational singularity in the same sense as a black hole. Black hole radiation couldn't have radiated away from it into spacetime, because there was no spacetime. Again, I'm not surprised you don't know this, because you don't even known who Stephen Hawking is.
I'm not going to give you a course in cosmology. You need to start reading basic articles until you have some rudimentary understanding. Until you do, attempting to teach you is a waste of time.
At first there was nothing and then it exploded is a pretty good tongue in cheek description of the big bang.
It's a pitiful straw man set up by people whose god is not the real God. It is wrong because: 1) there was no "at first" [time did not yet exist] and 2) there was no "explosion." So, on both ends of your supposedly "witty" description you're talking nonsense that has nothing to do with the vacuum, the initial singularity, and the inflation in which spacetime came into existence.
I don't need to tell you what existed "before" the singularity, because there was no "before." And I have bad news for you. If you're a Jew or a Christian, your theology agrees with this: God is outside of time.
As for the rest of it, in your pitiful version of cosmology, there was a Great Big Dude with a white beard who waved a magic wand over some water and created the earth before he created the sun, the moon and the stars. Unlike your description -- which has nothing to do with what physicists have put together for the beginning instant of time -- my tongue in cheek description is pretty much exactly what Genesis says.
And it's 100% baloney. It's a legend by a primitive tribe attempting to understand how things started, and not an actual description of what the real God did when the universe began.
I am pretty sure I never made such a claim, please correct me if I did. Waiting. I'd really like to debate your contention that something can spring into existence from nothing. I have real problems with that. E=MC^2, put a zero anywhere you like and the result is zero. So I am interested in hearing how zero turns into something other than zero.