  [Abbreviated] We investigate the dependence of Lyman-$\alpha$ emitter (LAE)
clustering on Lyman-$\alpha$ luminosity. We use 1030 LAEs from the MUSE-Wide
survey, 679 LAEs from MUSE-Deep, and 367 LAEs from the to-date deepest ever
spectroscopic survey, the MUSE Extremely Deep Field. All objects have
spectroscopic redshifts of $3<z<6$ and cover a large dynamic range of
Ly$\alpha$ luminosities: $40.15<\log (L_{\rm{Ly}\alpha}/[\rm{erg
\:s}^{-1}])<43.35$. We apply the Adelberger et al. K-estimator as the
clustering statistic and fit the measurements with state-of-the-art halo
occupation distribution (HOD) models. From the three main data sets, we find
that the large-scale bias factor, the minimum mass to host one central LAE,
$M_{\rm{min}}$, and (on average) one satellite LAE, $M_1$, increase weakly with
an increasing line luminosity. The satellite fractions are $\lesssim10$%
($\lesssim20$%) at $1\sigma$ ($3\sigma$) confidence level, supporting a
scenario in which DMHs typically host one single LAE. We next bisected the
three main samples into disjoint subsets to thoroughly explore the dependence
of the clustering properties on $L_{\rm{Ly}\alpha}$. We report a strong
($8\sigma$) clustering dependence on $L_{\rm{Ly}\alpha}$, where the highest
luminosity LAE subsample ($\log(L_{\rm{Ly}\alpha}/[\rm{erg
\:s}^{-1}])\approx42.53$) clusters more strongly
($b_{\rm{high}}=3.13^{+0.08}_{-0.15}$) and resides in more massive DMHs
($\log(M_{\rm{h}}/[h^{-1}\rm{M}_{\odot}])=11.43^{+0.04}_{-0.10}$) than the
lowest luminosity one ($\log(L_{\rm{Ly}\alpha}/[\rm{erg
\:s}^{-1}])\approx40.97$), which presents a bias of
$b_{\rm{low}}=1.79^{+0.08}_{-0.06}$ and occupies
$\log(M_{\rm{h}}/[h^{-1}\rm{M}_{\odot}])=10.00^{+0.12}_{-0.09}$ halos. We
discuss the implications of these results for evolving Ly$\alpha$ luminosity
functions, halo mass dependent Ly$\alpha$ escape fractions, and incomplete
reionization signatures.
