  We investigate the large $N$ instanton effects of partition functions in a
class of $\mathcal{N}=4$ circular quiver Chern-Simons theories on a
three-sphere. Our analysis is based on the supersymmetry localization and the
Fermi-gas formalism. The resulting matrix model can be regarded as a
two-parameter deformation of the ABJM matrix model, and has richer
non-perturbative structures. Based on a systematic semi-classical analysis, we
find analytic expressions of membrane instanton corrections. We also exactly
compute the partition function for various cases and find some exact forms of
worldsheet instanton corrections, which appear as quantum mechanical
non-perturbative corrections in the Fermi-gas system.
