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For comparison, if a bounded diagonal operator (or equivalently, a bounded degree-$d$ function on the Boolean cube) has $N(d)$ Pauli coefficients, all of magnitude $\\Omega(1/\\sqrt{N(d)})$, then $N(d)\\leq \\exp(\\widetilde{O}(d^{1.5}))$.\n  This construction implies the noncommutative Bohnenblust--Hille (BH) constant satisfies $\\mathrm{BH}_{M_2}(d)\\geq\\exp(\\Omega(d))$. 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For comparison, if a bounded diagonal operator (or equivalently, a bounded degree-$d$ function on the Boolean cube) has $N(d)$ Pauli coefficients, all of magnitude $\\Omega(1/\\sqrt{N(d)})$, then $N(d)\\leq \\exp(\\widetilde{O}(d^{1.5}))$.\n  This construction implies the noncommutative Bohnenblust--Hille (BH) constant satisfies $\\mathrm{BH}_{M_2}(d)\\geq\\exp(\\Omega(d))$. 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