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About the title, I didn't mean something like: Did you can... (because, of course we can use in the past since there's "did"). What I've known, could can also be the past simple form of can.

In my book, it said auxilary "can" used for impossibility (negative only) isn't changed and the only change is just the "be".

Example:

That can't be true! (Present/Future)

That can't have been true! (Past)

What about these two:

That couldn't be true!

That couldn't have been true!

Since the sentence isn't a request, I don't see the difference if the second examples is politer than the first examples. Also, I don't think it's about the time expression?

Can you tell me the difference or which one is correct if the others aren't?

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‘can't be’ means it's totally impossible, and ‘couldn't be’ means that it's almost impossible and it's often used in describing things happened in the past.

E.g.:If one said:

It couldn't have been an illusion.

He thought it was basically impossible that what he had just seen was a illusion.

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