  Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability
space $(X,{\mathcal B},\mu),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$
vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $\epsilon > 0$
and multicorrelation sequences of the form
$\displaystyle\alpha(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots
T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu$ we show that there exists a nilsequence
$\psi$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M}
\sum_{n=M}^{N-1} |\alpha(n) - \psi(n)| \leq \epsilon$ and $\displaystyle\lim_{N
\to \infty} \frac{1}{\pi(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |\alpha(p) -
\psi(p)| \leq \epsilon.$ This result simultaneously generalizes previous
results of Frantzikinakis [2] and the authors [11,13].
