What is Tsirelson's bound for the original bell's inequality As its well known, quantum correlations break Bell's inequalities only to a certain limit called the Tsirelson's bound. The bound was written for the CHSH inequality.
My question is what is the Tsirelson bound for the original case that Bell dealt with in his article, and how it is proven.
 A: The folklore says the upper bound is 3/2 but I don't remember a reference. I am going to reproduce some old notes of mine demonstrating that bound.
First, Bell's original inequality is eqn (15) of [1]:
$$1+P(\vec{b},\vec{c})\ge|P(\vec{a},\vec{b})-P(\vec{a},\vec{c})|,$$
where $a,b,c$ are unit vectors. The associated quantum prediction is obtained by using eqn (3) which states that
$$P(\vec{x},\vec{y})=-\vec{x}\cdot\vec{y}.$$
So the quantum version of the inequality can be written
$$\underbrace{|\vec{a}\cdot\vec{b}-\vec{a}\cdot\vec{c}|+\vec{b}\cdot\vec{c}}_{\displaystyle R}\le 1.$$
Your question would then be how badly $R$ can violate that upper bound of 1. By a suitable choice of basis, we can always write
$$\begin{align}
\vec{c}&=(\cos\varphi,\sin\varphi,0),\\
\vec{b}&=(\cos\varphi,-\sin\varphi,0),\\
\vec{a}&=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta),
\end{align}$$
with $\varphi,\phi\in[0,2\pi]$ and $\theta\in[0,\pi]$.
Then
$$\begin{align}
\vec{a}\cdot\vec{b}&=\cos(\phi+\varphi)\sin\theta,\\
\vec{a}\cdot\vec{c}&=\cos(\phi-\varphi)\sin\theta,\\
\vec{b}\cdot\vec{c}&=\cos^2\varphi-\sin^2\varphi,
\end{align}$$
and
$$R=2|\sin\varphi\sin\phi\sin\theta|+1-2\sin^2\varphi.$$
This is a second-order polynomial in $|\sin\varphi|$ whose maximum is reached for $|\sin\varphi|=\frac{1}{2}|\sin\phi\sin\theta|$ and the maximum value is $\frac{1}{2}|\sin\phi\sin\theta|+1.$
This latter bound is then maximum for $\phi=\theta=\pi/2$ and the maximum value is $3/2$. It is therefore reached for $\varphi=\pi/4$.
[1] John S. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1(3):195–200, 1964. (pdf)
A: Tsirelson's bound for CHSH works only with matrix sum. If one uses Kronecker sum instead then the eigenvalues are -4,-2,0,2,4. So the average of CHSH can reach 4 in tensor quantum mechanics.
