This answer to Major unsolved problems in theoretical computer science? question states that it is open if a particular problem in NP requires $\Omega(n^2)$ time.
Looking at the comments under answer made me wonder:
Aside from padding and similar tricks, what is the best known time complexity lower bound on a deterministic RAM machine (or multiple-tape deterministic Turing machine) for an interesting problem in NP (which is stated in a natural way)?
Is there any natural problem in NP which is known to be unsolvable in quadratic deterministic time on a reasonable machine model?
Essentially, what I am looking for is an example that rules out the following claim:
any natural NP problem can be solved in $O(n^2)$ time.
Do we know any NP problem similar to those in Karp's 1972 paper or Garey and Johnson 1979 that requires $\Omega(n^2)$ deterministic time? Or is it possible to the best of our knowledge that all interesting natural NP problems can be solved in $O(n^2)$ deterministic time?
Clarification to remove any confusion resulting from the mismatch between lower bound and not an upper bound: I am looking for a problem which we know we cannot solve in $o(n^2)$. If a problem satisfies the stronger requirement that $\Omega(n^2)$ or $\omega(n^2)$ time is needed (for all large enough inputs) then better, but infinitely often will do.