Did Snape know that Quirrell was possessed in The Philosopher's Stone?

Question

We know that Snape was tailing Quirrell throughout The Philosopher's Stone, trying to stop him from stealing the Stone. He did this on Dumbledore's orders. But is it possible to say what Snape's (and, by extention, Dumbledore's) motives were here? Was he aware of the bigger picture that Voldemort was trying to steal the Stone through Quirrell? Or was he unaware that Quirrell was possessed, and so thought that Quirrell was acting alone?

I know that he claimed to fellow-Death Eaters that he was unaware that Quirrell was possessed.

“I think you next wanted to know,” he pressed on, a little more loudly, for Bellatrix showed every sign of interrupting, “why I stood between the Dark Lord and the Philosopher's Stone. That is easily answered. He did not know whether he could trust me. He thought, like you, that I had turned from faithful Death Eater to Dumbledore’s stooge. He was in a pitiable condition, very weak, sharing the body of a mediocre wizard. He did not dare reveal himself to a former ally if that ally might turn him over to Dumbledore or the Ministry. I deeply regret that he did not trust me. He would have returned to power three years sooner. As it was, I saw only greedy and unworthy Quirrell attempting to steal the stone and, I admit, I did all I could to thwart him.”
_{(Half-Blood Prince, Chapter 2, Spinner's End).}

Just because Snape claims not to have known Voldemort was possessing Quirrell doesn't mean that he didn't actually know at the time. We know that he pulls the wool over the eyes of other Death Eaters all the time. This could be one such occasion. Claiming that he didn't know Voldemort was involved could just be part of his cover act.

He and Dumbledore would have had plenty of reasons to try and stop Quirrell, even if they didn't realise that he'd teamed up with Voldemort. And Snape suspected Quirrell for the whole year, even before Halloween.

Unfortunately, while everyone else was running around looking for it, Snape, who already suspected me, went straight to the third floor to head me off — and not only did my troll fail to beat you to death, that three-headed dog didn’t even manage to bite Snape’s leg off properly.
_{(Philosopher's Stone, Chapter 17, The Man With Two Faces).}

Harry makes the connection that Voldemort is involved after his encounter with the hooded Quirrell in the Forbidden Forest. Snape didn't benefit from this revelation, however, since Harry thought he was the one who was after the Stone.

Was Snape acting out of ignorance or a wider strategy? Did he think that he was stopping Voldemort or merely stopping Quirrell?

Answer 1

We're not sure how much Snape suspected.

Dumbledore certainly suspected Quirrell of something, and used Snape as an agent for keeping an eye on him. Snape was pretty smart, so he might have suspected.

Dumbledore turned a page, and said, without looking up, “Keep an eye on Quirrell, won’t you?”

From book 7, chapter 33, A Prince's Tale

Snape definitely did not see only greedy, unworthy Quirrell. In fact, he probably suspected something was up with Voldemort. However, he probably did not know that Voldemort was possessing him.

In muttering his countercurse at the Quidditch pitch and saving Harry, Snape shows that either he already suspected and was under orders to stop Quirrell OR he was simply doing it because of his obligation to protect Harry from his promises to Dumbledore and because James saved him once, or for Lily's memory. In this case he would immediately suspect that something was up with Quirrell.

In conclusion, he was certainly not just stopping Quirrell. He was following orders from Dumbledore. Whether he understood that Quirrell was an agent of Voldemort or not, we don't know. But he was just following Dumbledore's orders.

Answer 2

It’s unclear - he’d have acted the same whoever he thought it was.

We don’t know what Snape was thinking or suspected when Quirrell tried to steal the Philosopher’s Stone, nor can we deduce it from his actions. He was actually on Dumbledore’s side, so he’d try to stop Quirrell whether he knew it was actually the Dark Lord or not. All we know about what Snape was thinking when Quirrell tried to steal the Stone is what he told Bellatrix, which could very easily be a lie - he certainly wouldn’t have told her that he was knowingly working against the Dark Lord even though at least in other cases if not this one, he was. He certainly didn’t regret that the Dark Lord didn’t get the Stone and return to power, the rest of it could easily be a lie as well.

Snape does tell Quirrell to think about where his loyalties lie, but it’s unclear whether he’s referring to the Dark Lord, or if he meant it’d be disloyal to Dumbledore and Hogwarts to steal the Stone.

“Very well,’ Snape cut in. ‘We’ll have another little chat soon, when you’ve had time to think things over and decided where your loyalties lie.”
_{^{- Harry Potter and the Philosopher's Stone, Chapter 13 (Nicolas Flamel)}}

All we do know for sure is that he did attempt to stop Quirrell, and Dumbledore told him to watch Quirrell, so he was doing what Dumbledore had told him to do.

Answer 3

I think he knew, that Quirrell at the very least was a follower of the Dark Lord. While Snape's following remark could be interpreted as loyalty to Dumbledore, he had to have had suspicions after the Quidditch-game.

“Very well,’ Snape cut in. ‘We’ll have another little chat soon, when you’ve had time to think things over and decided where your loyalties lie.”

-Harry Potter and the Philosopher's Stone, Chapter 13 (Nicolas Flamel)

He countered Quirrell's curse aimed at Harry's broom. I strongly assume he knew, that it was Quirrell cursing him. What reason would a greedy, not concerned with anything dark- man have in doing that? At the very least, he hated Harry Potter, and since Harry was "the golden boy" in the eye of the public at that time and day, it was somebody who resented Harry Potter. This was in general the people who didn't adore him for killing the Dark Lord. The very in opposite in fact. I think from there it is not such a wide jump to suspect the Dark Lord himself involved. Especially with Dumbledore pointing out, that the Dark Lord isn't dead yet.