Proof.
By virtue of Proposition 
, the $\infty $-category $\operatorname{\mathcal{QC}}$ is generated under colimits by the object $\Delta ^1 \in \operatorname{\mathcal{QC}}$. The equivalence $(1) \Leftrightarrow (2)$ now follows by combining Remarks 9.3.1.17 and 5.5.4.5. We next show that, if condition $(2)$ is satisfied, then the core $\operatorname{\mathcal{C}}^{\simeq }$ is an $n$-truncated Kan complex. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms of $\operatorname{\mathcal{C}}$. Then the diagonal map
\[ \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Isom}(\operatorname{\mathcal{C}}) \quad \quad X \mapsto \operatorname{id}_{X} \]
is an equivalence of $\infty $-categories (Corollary 4.5.3.13), and therefore restricts to a homotopy equivalence of Kan complexes $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$. We are therefore reduced to showing that the Kan complex $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$ is $n$-truncated. Since $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$ is a summand of the Kan complex $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$, this follows immediately from assumption $(2)$ if $n \geq -1$. The case $n \leq -2$ then follows from the additional observation that if $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq }$ is nonempty, then the $\infty $-category $\operatorname{\mathcal{C}}$ is nonempty, so $\operatorname{Isom}(\operatorname{\mathcal{C}})^{\simeq }$ is nonempty.
We now complete the proof by showing that $(2)$ and $(3)$ are equivalent. By virtue of the preceding argument, we may assume that the core $\operatorname{\mathcal{C}}^{\simeq }$ is $n$-truncated, so the product $\operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq }$ is $n$-truncated (Remark 3.5.7.6). Using Proposition 3.5.9.13, we see that condition $(2)$ is satisfied if and only if the map of Kan complexes
\[ U: \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{\mathcal{C}}^{\simeq } \times \operatorname{\mathcal{C}}^{\simeq } \quad \quad (f: X \rightarrow Y) \mapsto (X,Y) \]
is $n$-truncated. Since $U$ is a Kan fibration (Corollary 4.4.5.4), this is equivalent to the requirement that each fiber of $U$ is an $n$-truncated Kan complex (Proposition 3.5.9.8), which is a restatement of $(3)$.
$\square$