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Higher order generalization of Cauchy-Schwarz?

Yes, because the OP stated that the ground field is $\mathbb{R}$, one can simply take the octic polynomial $$ Q(v_1,v_2,\ldots,v_n) = \sum_{1\le i < j\le n} \bigl((v_i,v_i)(v_j,v_j)-(v_i,v_j)^2\bigr)^ …
Robert Bryant's user avatar