function of the results of spin measurements on particles $1,2,\ldots,N$ along respective directions $\vec{e}_1,\vec{e}_2,\ldots,\vec{e}_N$ is given by $$\begin{aligned}
E_{12 \cdots N} \!\!\!&(&\!\!\!\vec{e}_1,\vec{e}_2,\ldots,\vec{e}_N)  \nonumber  \\
& = & \text{Tr}[\rho_{12\cdots N}
\sigma(\vec{e}_1)\otimes \sigma(\vec{e}_2)\otimes \cdots 
\otimes \sigma(\vec{e}_N)]   \nonumber  \\
& = & z_1 z_2 \cdots z_N \cos\vartheta_1 \cos\vartheta_2 \cdots \cos\vartheta_N ,\end{aligned}$$ which generalizes Eq. (9). Thus we have proved that, for the class of $n$-orthogonal states in Eq. (23), the unconditional $N$-particle correlations are compatible with local realism for any $N=2,3,\ldots,n-1$.

However, by adequately generalizing the CHSH inequality to an $N$-measurement scheme [@Hardy], one could equally prove [@Cereceda] that in order for the state (27) to yield the largest violation of the appropriate Bell inequality it is necessary that the $c$ numbers in front of $|z_1\rangle |z_2\rangle \cdots |z_N\rangle$ and $|-z_1\rangle |-z_2\rangle \cdots |-z_N\rangle$ have the same modulus.[^3] Obviously, when applied to the state (23), this condition means that $|c_1|=|c_2|=1/\sqrt{2}$. This in turn implies that all observers $N+1,N+2,\ldots,n$ must perform spin measurements within the respective $x$-$y$ plane (i.e., $\vartheta_j =\pi/2$ for all $j$) if the remaining state $|\Psi^{+}_{12\cdots N}\rangle$ for the other $N$ particles is to violate maximally Bell’s inequality. In any case, the amount of violation, if any, does depend on the 