We present detailed measurements and analysis of the guided acoustic wave Brillouin scattering (GAWBS)-induced depolarization noise in a multi-core fiber (MCF) used for a digital coherent optical transmission. We first describe the GAWBS-induced depolarization noise in an uncoupled four-core fiber (4CF) with a 125μm cladding and compare the depolarization noise spectrum with that of a standard single-mode fiber (SSMF). We found that off-center cores in the 4CF are dominantly affected by higher-order TR_n,m modes rather than the TR_2,m mode unlike in the center core, and the total power of the depolarization noise in the 4CF was almost the same as that in the SSMF. We also report measurement results for the GAWBS-induced depolarization noise in an uncoupled 19-core fiber with a 240μm cladding. The results indicate that the amounts of depolarization noise generated in the cores are almost identical. Finally, we evaluate the influence of GAWBS-induced polarization crosstalk (XT) on a coherent QAM transmission. We found that the XT limits the achievable multiplicity of the QAM signal to 64 in a transoceanic transmission with an MCF.