  We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are
the translation to the world of $n$-quasi-categories of groupoidal and
truncated $(\infty, n)$-$\Theta$-spaces defined by Rezk. We show that these
objects are the fibrant objects of model structures on the category of
presheaves on $\Theta_n$ obtained by localisation of Ara's model structure for
$n$-quasi-categories. Furthermore, we prove that the inclusion $\Delta \to
\Theta_n$ induces a Quillen equivalence between the model structure for
groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen
model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To
get to these results, we also construct a cylinder object for
$n$-quasi-categories.
