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The purpose of this article is threefold. First, we show that when oneexplores a conformal loop ensemble of parameter $\kappa=4$ ($\mathrm{CLE}_4$)on an independent $2$-Liouville quantum gravity ($2$-LQG) disk, the surfaceswhich are cut out are independent quantum disks. To achieve this, we rely onapproximations of the explorations of a $\mathrm{CLE}_4$: we first approximatethe $\mathrm{SLE}_4^{\langle \mu \rangle}$ explorations for $\mu \in\mathbb{R}$ using explorations of the $\mathrm{CLE}_\kappa$ as $\kappa \uparrow4$ and then we approximate the uniform exploration by letting $\mu \to \infty$.Second, we describe the relation between the so-called natural quantum distanceand the conformally invariant distance to the boundary introduced by Werner andWu. Third, we establish the scaling limit of the distances from the boundary tothe large faces of $3/2$-stable maps and relate the limit to the$\mathrm{CLE}_4$-decorated $2$-LQG.

Distances on the $\mathrm{CLE}_4$, critical Liouville quantum gravity and $3/2$-stable maps