  We prove that SLE$_\kappa$ for $\kappa \in (4,8)$ on an independent
$\gamma=4/\sqrt{\kappa}$-Liouville quantum gravity (LQG) surface is uniquely
characterized by the form of its LQG boundary length process and the form of
the conditional law of the unexplored quantum surface given the explored
curve-decorated quantum surface up to each time $t$. We prove variants of this
characterization for both whole-plane space-filling SLE$_\kappa$ on an
infinite-volume LQG surface and for chordal SLE$_\kappa$ on a finite-volume LQG
surface with boundary. Using the equivalence of Brownian and $\sqrt{8/3}$-LQG
surfaces, we deduce that SLE$_6$ on the Brownian disk is uniquely characterized
by the form of its boundary length process and that the complementary connected
components of the curve up to each time $t$ are themselves conditionally
independent Brownian disks given this boundary length process.
  The results of this paper are used in another paper by the same authors to
show that the scaling limit of percolation on random quadrangulations is given
by SLE$_6$ on $\sqrt{8/3}$-LQG with respect to the
Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the
Gromov-Hausdorff topology for curve-decorated metric measure spaces.
