Existing tests for high-dimensional white noise are tailored to the two ends of the sparsity spectrum: maximum-type statistics excel against a few dominant correlations and sum-type statistics against many weak ones, leaving the intermediate regime without a systematic tool. We propose a family of $U$-statistics built from the entry-wise $\ell_a$-norms of the sample autocovariance matrices, in which the even order $a$ interpolates between the sum-type ($a=2$) and maximum-type ($a\to\infty$) endpoints. As $p$ and $T$ diverge jointly, martingale difference theory yields joint asymptotic normality across orders together with asymptotic independence, so that several orders can be combined after a simple calibration. Neither cross-sectional independence nor a mixing condition is required, only a spectral condition on $\bSig_0$; the derivation links cross-sectional correlation to a graph structure that prices each moment edge by edge. Simulations confirm accurate size and power that is stable across sparse, intermediate and dense alternatives, and show that the low-order members of the family retain their level under conditional heteroscedasticity, where the existing tests distort severely.

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