Projective Hilbert space
Two state vectors differing only by a nonzero complex factor represent the same pure quantum state.
$$\mathbb{CP}^{n-1}=\big(\mathbb C^n\setminus\{0\}\big)/\mathbb C^*$$
What this means
Two state vectors differing only by a nonzero complex factor represent the same pure quantum state.
Deep dive: mathematical and physical mechanism
Normalized rays form complex projective space. For a qubit, $\mathbb{CP}^1\cong S^2$, the Bloch sphere. This is why a qubit can exhibit geometric phases associated with paths on a sphere.
How it appears in quantum phenomena
Qubit control pulses trace paths on the Bloch sphere.
Assumptions, caveats and limitations
The Bloch sphere describes pure single-qubit rays, not arbitrary multi-qubit states.
Connect the ideas
For context, see the Geometric Quantum Mechanics topic guide and the reading references.