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Motivated by applications in genomics, we study in this paper four interre-
lated high-dimensional hypothesis testing problems on dependence structures among
multiple populations. A new test statistic is constructed for testing the global hypoth-
esis that multiple covariance matrices are equal, and its limiting null distribution is
established. Correction methods are introduced to improve the accuracy of the test
for finite samples. It is shown that the proposed tests are powerful against sparse
alternatives and enjoy certain optimality properties. We then propose a multiple test-
ing procedure for simultaneously testing the equality of the entries of the covariance
matrices across multiple populations. The proposed method is shown to control the
false discovery rate. A simulation study demonstrates that the proposed tests maintain
the desired error rates under the null and have good power under the alternative. The
methods are also applied to a Novartis multi-tissue analysis. In addition, testing and
support recovery of submatrices of multiple covariance matrices are studied.