Bloch sphere
The Bloch sphere is a picture that lets you see the state of a single spin-½ particle, or any two-level quantum system, as a point on the surface of an ordinary ball. The north pole is spin up, the south pole is spin down, and every other point on the surface stands for some superposition of the two. Two complex numbers that would be hard to picture become a simple arrow pointing somewhere on a globe.
Points on the equator are equal-weight superpositions of up and down, differing only in their phase, which is read off as the angle around the equator. This turns the abstract phase between two amplitudes into a visible direction. A spin actually pointing along some compass direction in space sits at exactly the matching point on the sphere, so the picture is wonderfully intuitive.
On the Bloch sphere, operations on a spin become rotations of the sphere, which is why it is the everyday workhorse of quantum computing. A magnetic field makes the arrow precess as a steady rotation about an axis; a quantum gate is a precise rotation by a chosen angle. Pure states live on the surface, while the inside represents mixed states with some classical uncertainty, so the whole ball maps the full range of a qubit.
Two angles, a polar tilt θ and an azimuth φ, place any pure spin state on the sphere.
The angle θ on the sphere is twice the real physical angle: states a full 180° apart on the sphere, like up and down, are genuinely opposite, while a 90° tilt on the sphere corresponds to a 45° rotation in space.