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In this review paper I present two geometric constructions of distinguishednature, one is over the field of complex numbers $\mathbb{C}$ and the other oneis over the two elements field $\mathbb{F}_2$. Both constructions have beenemployed in the past fifteen years to describe two quantum paradoxes or tworesources of quantum information: entanglement of pure multipartite systems onone side and contextuality on the other. Both geometric constructions arelinked to representation of semi-simple Lie groups/algebras. To emphasize thisaspect one explains on one hand how well-known results in representation theoryallows one to see all the classification of entanglement classes of varioustripartite quantum systems ($3$ qubits, $3$ fermions, $3$ bosonic qubits...) ina unified picture. On the other hand, one also shows how some weight diagramsof simple Lie groups are encapsulated in the geometry which deals with thecommutation relations of the generalized $N$-Pauli group.

Geometric constructions over $\mathbb{C}$ and $\mathbb{F}_2$ for Quantum Information