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We study a mean change point testing problem for high-dimensional data, withexponentially- or polynomially-decaying tails. In each case, depending on the$\ell_0$-norm of the mean change vector, we separately consider dense andsparse regimes. We characterise the boundary between the dense and sparseregimes under the above two tail conditions for the first time in the changepoint literature and propose novel testing procedures that attain optimal ratesin each of the four regimes up to a poly-iterated logarithmic factor. Bycomparing with previous results under Gaussian assumptions, our resultsquantify the costs of heavy-tailedness on the fundamental difficulty of changepoint testing problems for high-dimensional data.  To be specific, when the error vectors follow sub-Weibull distributions, aCUSUM-type statistic is shown to achieve a minimax testing rate up to$\sqrt{\log\log(8n)}$. When the error distributions have polynomially-decayingtails, admitting bounded $\alpha$-th moments for some $\alpha \geq 4$, weintroduce a median-of-means-type test statistic that achieves a near-optimaltesting rate in both dense and sparse regimes. In particular, in the sparseregime, we further propose a computationally-efficient test to achieve theexact optimality. Surprisingly, our investigation in the even more challengingcase of $2 \leq \alpha < 4$, unveils a new phenomenon that the minimax testingrate has no sparse regime, i.e.\ testing sparse changes isinformation-theoretically as hard as testing dense changes. This phenomenonimplies a phase transition of the minimax testing rates at $\alpha = 4$.

Robust mean change point testing in high-dimensional data with heavy tails