  In a calculation of rotated matrix elements with angular momentum projection,
the generalized Wick's theorem may encounter a practical problem of
combinatorial complexity when the configurations have more than four
quasi-particles (qps). The problem can be solved by employing the Pfaffian
algorithm generally applicable to calculations of matrix elements for
Hartree-Fock-Bogoliubov states with any number of qps. This breakthrough in
many-body techniques enables studies of high-spin states in a shell-model
framework. As the first application of the Pfaffian algorithm, the
configuration space of the Projected Shell Model is expanded to include 6-qp
states for both positive and negative parities. Taking $^{166}$Hf as an
example, we show that 6-qp states become the main configuration of the yrast
band beyond spin $I \approx 34\hbar$, which explains the observed third
back-bending in moment of inertia. Structures of multi-qp high-$K$ isomers in
$^{176}$Hf are analyzed as another example.
