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In this thesis we use the language of sheaf theory in order to develop adeeper understanding of some of the fundamental differences - such asentanglement, contextuality and non-locality - between quantum and classicalphysics. We first present, based on the work of Abramsky and Brandenburger, howsheaves, defined over certain posets of physically meaningful contexts, give anatural setting for capturing and analysing important quantum mechanicalphenomena, such as quantum non-locality and contextuality. We also describe howthis setting naturally leads to a three level hierarchy of quantumcontextuality: weak contextuality, logical non-locality and strongcontextuality. We use these insights in order to classify certain multipartiteentangled states. Almost all of these turn out to be at least logicallynon-local, and a number of them even turn out to be strongly contextual. Wefurther extend this result by showing that all n-qubit entangled states, withthe exception of tensor products of single-qubit and bipartitemaximally-entangled states, are logically non-local. Our proof is constructive:given any n-qubit state, we present an algorithm which produces n+2 localobservables witnessing its logical non-locality. In the second half of thethesis we use the same basic principle of sheaves defined over physicallymeaningful contexts, in order to present an elegant mathematical language,known under the name of the Topos Approach, in which many quantum mechanicalconcepts, such as states, observables, and propositions about these, can beexpressed. We then show that the language of the Topos Approach is as least asexpressive, in logical terms, as traditional quantum logic. Finally, startingfrom a topos-theoretic perspective, we develop the construction of contextualentropy in order to give a unified treatment of classical and quantum notionsof information theoretic entropy.

Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory