  We conducted a systematic survey of the environment of high-z quasars at
submillimeter wavelengths to unveil and characterize the surrounding
distribution of dusty submillimeter galaxies (SMGs). We took sensitive
JCMT/SCUBA-2 observations for 3 enormous Lyman-alpha nebulae (ELANe) and 17
quasar fields in the redshift range 2<z<4.2 selected from recent Ly$\alpha$
surveys. These observations uncovered 523 and 101 sources at 850$\mu$m and
450$\mu$m, respectively, with S/N>4 or detected in both bands at S/N>3. We ran
Monte Carlo simulations to construct 850$\mu$m number counts and unveil an
excess of sources in 75% of the targeted fields. Overall, regions around ELANe
and quasars are overabundant with respect to blank fields by a factor of
$3.4\pm0.4$ and $2.5\pm0.2$, respectively. Therefore, the excess of SMGs is
likely part of the Mpc-scale environment around these systems. By combining all
fields and repeating the count analysis in radial apertures, we find a decrease
in the overdensity factor from >3 within $\sim 2$ cMpc to $\sim2$ at the edge
of the surveyed field ($\sim10$ cMpc), suggesting that the physical extent of
the overdensities is larger than our maps. We computed preferred directions for
the overdensities of SMGs from the positions of the sources and used them to
orient and create stacked maps of source densities for the quasars'
environment. This stacking unveils an elongated structure reminiscent of a
large-scale filament with a scale width of $\approx 3$ cMpc. Finally, the
directions of the overdensities are roughly aligned with the major axis of the
Ly$\alpha$ nebulae, suggesting that the latter trace, on hundreds of kpc, the
central regions of the projected large-scale structure described by the SMGs on
Mpc scales. Confirming member associations of the SMGs is required to further
characterize their spatial and kinematic distribution around ELANe and quasars.
