  We study the problem of community recovery and detection in multi-layer
stochastic block models, focusing on the critical network density threshold for
consistent community structure inference. Using a prototypical two-block model,
we reveal a computational barrier for such multi-layer stochastic block models
that does not exist for its single-layer counterpart: When there are no
computational constraints, the density threshold depends linearly on the number
of layers. However, when restricted to polynomial-time algorithms, the density
threshold scales with the square root of the number of layers, assuming
correctness of a low-degree polynomial hardness conjecture. Our results provide
a nearly complete picture of the optimal inference in multiple-layer stochastic
block models and partially settle the open question in Lei and Lin (2022)
regarding the optimality of the bias-adjusted spectral method.
