Why was Snape arguing with Quirrell?

It first appears that Snape is bullying Quirrell—who appears to be stuttering, timid man—into helping him steal the Philosopher’s Stone. How much Snape knew or suspected about Quirrell working for Voldemort was deliberately unclear, in order of obfuscate Snape’s true allegiance.

Why does Snape threaten Quirrell?

For this particular one it is because Snape didn’t know that it was Voldemort under the turban, so he thought Professor Quirrel was simply going after the stone to get the glory for himself.

Why did Voldemort allow Snape to stay with Dumbledore?

Perhaps Voldemort thought Snape was biding his time till the Dark Lord rose once again (while also protecting the stone from another professor whom Snape assumed was attempting to steal the stone). As long as Voldemort remained hidden, he knew it was best for Snape to appear loyal to Dumbledore in anyway possible.

Why did Voldemort keep the Quirrell stone?

Voldemort must have thought that Snape didn’t know Voldemort was inside the body of Quirrell, He just had to keep the stone safe, because he wouldn’t get trusted by dumbledore.

Why didn’t Snape turn to Dumbledore for the Quirrell stone?

Ultimately though, had Snape, from Voldemort’s perspective, truly considered Quirrell an objective danger to Dumbledore’s plans for the stone, Snape would have turned to Dumbledore. Instead, Snape seems to be pursuing this on his own, which logically means Snape isn’t looking out for Dumbledore’s plans, but his (Snape’s) own plans.

Why didn’t Pettigrew tell Voldemort about Lily Potter?

And Pettigrew almost certainly couldn’t have told Voldemort, because there’s no reason to believe he ever knew. As soon as Dumbledore found out about the Potters being targeted, he used Snape’s feelings for Lily to make him turn spy against Voldemort, which was a very valuable asset to him.