Large violations in Kochen Specker contextuality and their applications

The Kochen-Specker (KS) theorem is a fundamental result in quantum foundations that has spawned massive interest since its inception. We present state-independent non-contextuality inequalities with large violations, particular, we exploit connection between proofs and pseudo-telepathy games to show KS Hilbert spaces of dimension $d \geq 2^{17}$ the ratio value classical bias being $O(\sqrt{d}/\log d)$. study properties this set applications violation. It been recently shown always consist substructures state-dependent contextuality called $01$-gadgets or bugs. one-to-one $\mathbb{C}^d$ Hardy paradoxes for maximally entangled state $\mathbb{C}^d \otimes \mathbb{C}^d$. use construct violation arbitrary vectors $\mathbb{C}^d$, as well novel \mathbb{C}^d$, give these constructions. As technical result, minimum faithful orthogonal representation graph $\mathbb{R}^d$ not monotone, may be independent interest.