  We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete
cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples
$(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with
$a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$,
and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if
$d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube
$\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples
$(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with
$a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$) is at most $|A|^{\log_2{
\binom{2k}{k}}}$, and $\log_2{ \binom{2k}{k}}$ is the best possible exponent.
We discuss the analogous problems for the sets $\{0,1,\dots,n\}^d$ for $n\geq
2$.
