  It is a well established fact that the notions of quasi-abelian categories
and tilting torsion pairs are equivalent. This equivalence fits in a wider
picture including tilting pairs of $t$-structures.
  Firstly, we extend this picture into a hierarchy of $n$-quasi-abelian
categories and $n$-tilting torsion classes. We prove that any $n$-quasi-abelian
category admits a derived category endowed with a $n$-tilting pair of
$t$-structures such that the respective hearts are derived equivalent.
  Secondly, we describe the hearts of these $t$-structures as quotient
categories of coherent functors, generalizing Auslander's Formula.
  Thirdly, we apply our results to Bridgeland's theory of perverse coherent
sheaves for flop contractions. In Bridgeland's work, the relative dimension $1$
assumption guaranteed that $f_*$-acyclic coherent sheaves form a $1$-tilting
torsion class, whose associated heart is derived equivalent to $D(Y)$. We
generalize this theorem to relative dimension $2$.
