  Mergers of black holes (BHs) and neutron stars (NSs) result in the emission
of gravitational waves that can be detected by LIGO. In this paper, we look at
2+2 and 3+1 quadruple-star systems, which are common among massive stars, the
progenitors of BHs and NSs. We carry out a detailed population synthesis of
quadruple systems using the MSE code, which seamlessly takes into consideration
stellar evolution, binary and tertiary interactions, $N$-body dynamics, and
secular evolution. We find that, although secular evolution plays a role in
compact object (BH and NS) mergers, (70--85) \% (depending on the model
assumptions) of the mergers are solely due to common envelope (CE) evolution.
Significant eccentricities in the LIGO band (higher than 0.01) are only
obtained with zero supernova (SNe) kicks and are directly linked to the role of
secular evolution. A similar outlier effect is seen in the
$\chi_{\mathrm{eff}}$ distribution, with negative values obtained only with
zero SNe kicks. When kicks are taken into account, there are no systems that
evolve into a quadruple consisting of four compact objects. For our fiducial
model, we estimate the merger rates (in units of $\Gpcyr$) in 2+2 quadruples
(3+1 quadruples) to be 10.8 $\pm$ 0.9 (2.9 $\pm$ 0.5), 5.7 $\pm$ 0.6 (1.4 $\pm$
0.4) and 0.6 $\pm$ 0.2 (0.7 $\pm$ 0.3) for BH-BH, BH-NS and NS-NS mergers
respectively. The BH-BH merger rates represent a significant fraction of the
current LIGO rates, whereas the other merger rates fall short of LIGO
estimates.
