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We prove two types of optimal dimension-free $L^p$ inequalities,$1<p<\infty$, for low and high degree scalar-valued functions on the Hammingcube. More precisely, any function $f$ with spectrum bounded from above by $d$satisfies the Bernstein--Markov type inequality  $$ \|\Delta^k f\|_{L^p(\{-1,1\}^n)} \le Q(f)^\varepsilon C(p,\varepsilon)^k\,d^k\, \|f\|_{L^p(\{-1,1\}^n)},$$ for any $\varepsilon>0$ and $k\in\mathbb N$,and any function $f$ with spectrum bounded from below by $d$ satisfies theupper bound on the decay of the heat semigroup  $$ \|e^{-t\Delta}f\|_{L^p(\{-1,1\}^n)} \leQ(f)^\varepsilon\exp(-c(p,\varepsilon) td) \|f\|_{L^p(\{-1,1\}^n)},$$ for any$t, \varepsilon>0$. Here, $Q(f)$ denotes the quotient between the largest andsmallest non-zero values of $f$, and the constants $c(p,\varepsilon)$ and$C(p,\varepsilon)$ depend only on $p$ and $\varepsilon$; crucially, they areindependent of the dimension $n$. In particular, if $f$ is Boolean-valued or$f\colon \{-1,1\}^n\to \{-1,0,1\},$ then the bounds we obtain are independentof $Q(f)$. We also provide counterexamples at the endpoint $p=\infty$, forwhich a linear growth in $d$ does not suffice when $k=1$.

Dimension-free inequalities for low and high degree functions on the Hamming cube