Understanding kQBF: changing order of quantification?
$\exists x\forall y \hspace{1mm}\varphi$ is not equivalent to $\forall y \exists x \hspace{1mm} \varphi$.

Consider for example the formula "for all $x$ there exists $y$ such that $y=x^2$". When evaluated over the reals, this is true (every real number has a square). However, if you switch the quantifiers you get the statement "there exists $y$ such that for all $x$ it holds that $y=x^2$". Note that this has an entirely different meaning, since now the statement reads "there exists a real number which is the square of all real numbers", and this is obviously false.

The order of quantifiers is important, so you can't trivially reduce $kQBF$ to $(k-1)QBF$.