For now I will only give you an overview of the ideas involved and show you how you should interpret the idea of a "local realistic theory" that cannot exist at the microscopic scale. Once you've read it, and if you feel you need more mathematical rigor to be convinced, then I will draw you step by step the proof of Bell's inequality (it is not the only one leading to the same claims, just one of the first ones that did so), as it is rather neat.
Einstein reality: Property of system already determined before the measurement. Meaning the system "has them".
Einstein locality: Physical reality described in a local manner. Independently of measurements that are carried out on spatially separated systems: "no action at a distance".
Now Bell's inequality showed that for entangled system, Einstein's description (or maybe expectation) of reality/locality of physical systems are both undermined.
Bell's approach:
Assuming that each photon carries a hidden variable $\lambda$ that determines the outcome of polarization experiments at A and B for any angles of the polarimeters $\delta_1$ and $\delta_2$:
$$\begin{align}
S_A^{\lambda}(\delta_1) = {+1,-1}\\
S_B^{\lambda}(\delta_2) = {+1,-1}
\end{align}
$$
The two $S$ functions contain the possible outcomes of a polarization measurement (for each system as detailed in the equation), and the result already defined (either -1 or +1) because $S$ is dependent on a hidden variable $\lambda$ providing the measurement outcome before it has taken place.
The variable $\lambda$ itself has a probability density distribution as follows:
$$ \rho (\lambda) \ge 0, \int \rho(\lambda)d\lambda=1$$
Now using the classical correlation coefficient (product of $S_A$ and $S_B$ expresses locality):
$$\epsilon^{cl}(\delta_1,\delta_2)= \int \rho(\lambda)S_A^{\lambda}(\delta_1)S_B^{\lambda}(\delta_2)d\lambda$$
From this equation, Bell derived his famous inequality (proof of which I was referring to at the start):
$$\left|\epsilon^{cl}(\delta_1,\delta_2) - \epsilon^{cl}(\delta_1,\delta_3)\right| \leq 1-\epsilon^{cl}(\delta_2,\delta_3) $$
Having now all the necessary ingredients, the next step is to measure correlation coefficients at different angles $\delta_1, \delta_2, \delta_3$, and see if Bell's inequation holds or not: (if it holds then Einstein's views would have been plausible)
Now by choosing: $\delta_1=30°, \delta_2=60°, \delta_3=90°$
Calculating the correlation coefficients in quantum mechanics and then compared.
First definition of correlation in quantum mechanics:
$$\begin{align}
\epsilon^{AB}(\alpha,\beta) :&= \left<\Phi_+^{AB}\right. \left|E^{A}(\alpha)\otimes E^{B}(\beta) \right| \left. \Phi_+^{AB}\right> \\
\epsilon^{AB}(\alpha,\beta) &= P_{++}+P_{--}-P_{+-}-P_{-+} \\
\epsilon^{AB}(\alpha,\beta) &= \cos2(\beta-\alpha)
\end{align}$$
The above are the generalized formulas, where $\alpha$ and $\beta$ are the polarimeter angles, $E^{A}$ and $E^{B}$ are the polarization operators of system of photon A and system of photon B respectively, $\Phi_+^{AB}$ (entangled state of choice) is one of the 4 Bell states (for 2 particle systems) and $P_{++},...$ are the probabilities to measure both polarization as horizontal, $--$ for measuring both vertical polarizations and so on. To reach the simplified formula with $\cos$, just calculate each term in the second equation (using the first equation).
Back to our measurements, now using $\epsilon^{AB}(\alpha,\beta) = \cos2(\beta-\alpha)$ we have:
$$\epsilon^{AB}(\delta_1,\delta_2)=\frac{1}{2}, \epsilon^{AB}(\delta_1,\delta_3)=-\frac{1}{2}, \epsilon^{AB}(\delta_2,\delta_3)=\frac{1}{2} $$
Inserting the results back into Bell's inequality yields: $1 \leq \frac{1}{2}$
It is clear that Bell's inequality is violated using the quantum mechanical definition of the correlation coefficient, meaning the Quantum theory and local-realistic theories leads to contradictory results.
To sum up, it was shown that there cannot be a so called "hidden" variable for each measurement that would predict the outcome before it is actually performed. Which brings us to the correct assessment of entangled states which is:
"The quantum state of each particle cannot be described independently, and measurements can be correlated even if the two entangled systems are light years apart."
