High-dimensional longitudinal data such as time-course microarray data are now widely available. One important feature of such data is that, for each individual, high-dimensional measurements are repeatedly collected over time. Moreover, these measurements are spatially and temporally dependent which, respectively, refers to dependence within each particular time point and among different time points. This paper focuses on testing the homogeneity of covariance matrices of high-dimensional measurements over time against the change-point type alternatives. We allow the dimension of measurements (p) to be much larger than the number of individuals (n). Specifically, a test statistic for the equivalence of covariance matrices is proposed and the asymptotic normality is established. In addition to testing, an estimator for the location of the change point is given whose rate of convergence is established and shown to depend on p, n and the signal-to-noise ratio. The proposed method is extended to locate multiple change points by applying a binary segmentation approach, which is shown to be consistent under some mild conditions. The proposed testing procedure and change-point identification methods are able to accommodate both spatial and temporal dependences. Simulation studies and an application to a time-course microarray data set are presented to demonstrate the performance of the proposed method.