  The subject of this paper is a nerve construction for bicategories introduced
by Leinster, which defines a fully faithful functor from the category of
bicategories and normal pseudofunctors to the category of presheaves over
Joyal's category $\Theta_2$. We prove that the nerve of a bicategory is a
$2$-quasi-category (a model for $(\infty,2)$-categories due to Ara), and
moreover that the nerve functor restricts to the right part of a Quillen
equivalence between Lack's model structure for bicategories and a Bousfield
localisation of Ara's model structure for $2$-quasi-categories. We deduce that
Lack's model structure for bicategories is Quillen equivalent to Rezk's model
structure for $(2,2)$-$\Theta$-spaces on the category of simplicial presheaves
over $\Theta_2$.
  To this end, we construct the homotopy bicategory of a $2$-quasi-category,
and prove that a morphism of $2$-quasi-categories is an equivalence if and only
if it is essentially surjective on objects and fully faithful. We also prove a
Quillen equivalence between Ara's model structure for $2$-quasi-categories and
the Hirschowitz--Simpson--Pellissier model structure for
quasi-category-enriched Segal categories, from which we deduce a few more
results about $2$-quasi-categories, including a conjecture of Ara concerning
weak equivalences of $2$-categories.
