Let $\Gamma$ be a finitely generated group, and let $\mathrm{Rep}(\Gamma, \mathrm{SO}(2, n))$ be the moduli space of representations of $\Gamma$ into $\mathrm{SO}(2, n) (n \geq 2)$. An element $\rho : \Gamma \to \mathrm{SO}(2, n)$ of $\mathrm{Rep}(\Gamma , \mathrm{SO}(2, n))$ is quasi-Fuchsian if it is faithful, discrete, preserves an acausal $(n-1)$-sphere in the conformal boundary $\mathrm{Ein}_n$ of the anti-de Sitter space, and if the associated globally hyperbolic anti-de Sitter space is spatially compact—a particular case is the case of Fuchsian representations, i.e., composition of a faithful, discrete, and cocompact representation $\rho_f : \Gamma \to \mathrm{SO}(1, n)$ and the inclusion $\mathrm{SO}(1, n) \subset \mathrm{SO}(2, n)$.

In *Anosov AdS representations are quasi-Fuchsian* we proved that quasi-Fuchsian representations are precisely representations that are Anosov as defined in *Anosov flows, surface groups and curves in projective space*. In the present paper, we prove that the space of quasi-Fuchsian representations is open and closed, i.e., that it is a union of connected components of $\mathrm{Rep}(\Gamma, \mathrm{SO}(2, n))$.

The proof involves the following fundamental result: Let $\Gamma$ be the fundamental group of a globally hyperbolic spatially compact spacetime locally modeled on $\mathrm{AdS}_n$, and let $\rho : \Gamma \to SO_0(2, n)$ be the holonomy representation. Then, if $\Gamma$ is Gromov hyperbolic, the $\rho (\Gamma)$-invariant achronal limit set in $\mathrm{Ein}_n$ is acausal.

Finally, we also provide the following characterization of representations with zero-bounded Euler class: they are precisely the representations preserving a closed achronal subset of $\mathrm{Ein}_n$.