Is there an explanation for the difficulty of proving quadratic lower bounds for interesting NP problems?
Here is an explanation from Lipton's blog: Bait and Switch: Why Are Lower Bounds so Hard? 

As Grochow observed, Rudich's insight applies equally well to $n^2$ lower bounds as to super-polynomial lower bounds. 

Rudich's insight explains why any lower bound proof that it is based on the following method cannot work. 


  "any computation that computes $f$ must make slow progress toward $f$. Each computational step can only get you a little bit closer to the final goal, thus the computation will take many steps."


Basically, there is no measure of progress that can survive the Bait and Switch trick of Rudich and successfully leads to a lower bound.