Best $\Pi_k \text{SAT}$ running time?
These questions are discussed in detail in the paper


  Rahul Santhanam, Ryan Williams:
  "Beating Exhaustive Search for Quantified Boolean Formulas and Connections to Circuit Complexity"
  Electronic Colloquium on Computational Complexity (ECCC) 2013, #108
  https://eccc.weizmann.ac.il/report/2013/108/