As I recall, after the first book, Professor Quirrell is never even as much as mentioned until Order of the Phoenix. At the end of the first book, we were told that the events surrounding Harry getting the stone are a secret. However, it seems strange to me that a man's unexpected death - by the hands of one of his students in a forbidden corridor - never got any official questioning. My suspicion peaked during chapter 15 of Order of the Phoenix, where Harry declares to Umbridge, a ministry official, that:
“Yeah, Quirrell was a great teacher,” said Harry loudly, “there was just that minor drawback of him having Lord Voldemort sticking out of the back of his head.”
Obviously, this is a big deal. Is this knowledge that the ministry has? Was there any official explanation at all? If the ministry does know about this, then it's either a very inconvenient fact or yet another thing that they should be (at least trying to be) getting Harry in trouble for.