Sufficient conditions for the collapse of Polynomial Hierarchy (PH)
There are a (growing) number of parameterized complexity results where the existence of a polynomial-sized kernelization implies the collapse of the PH to the third level. The central technique is given in [1], building on prior work (referenced in [1]).

As a simple example, the $k$-Path problem is the parameterized version of the Longest Path problem:


  $k$-Path
  Instance: A graph $G$ and integer $k$.
  Parameter: $k$.
  Question: Does $G$ contain a path of length $k$?


This problem is in FPT (with somewhat practical algorithms), but in [2] they show that if it has a polynomially sized kernel (in $k$), then the PH collapses to $\Sigma^{P}_{3}$. (The current presentation is typically phrased as a negative kernalization result unless NP $\subseteq$ coNP/poly or coNP $\subseteq$ NP/poly, so searching for something like "no polynomial kernel unless" nets a lot of results.)

References


H. L. Bodlaender, B. M. P. Jansen, and S. Kratsch, "Kernelization lower bounds by cross-composition", SIAM J. Discrete Math., 28 (2014), pp. 277–305. [arXiv version]
H. L. Bodlaender, R. G. Downey, M. R. Fellows, D. Hermelin, "On problems without polynomial kernels", Journal of Computer and System Sciences, 75(8):423-434. 2009. [Stanford hosted version]