In the proof of Theorem 1 in [this paper](http://drops.dagstuhl.de/opus/volltexte/2019/10852/pdf/LIPIcs-CCC-2019-30.pdf) by Chen, McKay, Murray, and Williams the authors assume $\mathsf{NP} \subseteq \mathsf{P}/\mathsf{poly}$ and (in different parts of the proof) state this implies the following two inclusions:
 
 1. $\Sigma_3 \mathsf{TIME}(n^c) \subseteq \mathsf{SIZE}(n^{O(c)})$ for every $c$
 2. $\mathsf{ZPP}^\mathsf{NP} \subseteq \mathsf{P}/\mathsf{poly}$

They also cite [this](https://epubs.siam.org/doi/pdf/10.1137/S0097539795296206) enhancement of the Karp-Lipton theorem, in which the collapse of $\mathsf{PH}$ is to $\mathsf{ZPP}^\mathsf{NP}$.
I suspect the theorem is behind the inclusions in some way, but I just can't make the connection.

What am I missing?