local hidden variables
A local hidden-variable theory is an attempt to explain quantum randomness by supposing that particles carry extra, unseen properties that fix their measurement outcomes in advance. The 'hidden' part means these properties are not in the standard wavefunction; the 'local' part means each particle's hidden instructions are set where and when it was prepared, and are not changed by anything done to a distant partner. It is the intuitive picture EPR hoped would complete quantum mechanics.
On this view, an entangled pair would simply be two particles that left the source already agreeing on how to answer every possible question — like two travellers who synchronised their decisions before parting. Their later correlations would be no more mysterious than two gloves from the same pair: open one box and find a left glove, and you instantly 'know' the other is right, with nothing spooky going on.
The decisive blow is that this comforting story is false. Bell showed that any such locally-predetermined scheme obeys an inequality, and experiments find quantum systems break it. The glove analogy fails because real entangled particles answer questions that were not even decided in advance, and they do so with correlations too strong for any shared prior agreement. Nature has firmly ruled out local hidden variables as the explanation.
Each result depends only on the local setting and a shared hidden λ — a scheme Bell tests rule out.
Ruling out local hidden variables does not rule out all hidden variables. Bohmian mechanics reproduces quantum predictions with hidden positions, but only by being explicitly non-local — keeping the variables while sacrificing locality.