When is a pure state of three qubits determined by its single-particle reduced density matrices?

Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so-called Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecision, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger-Horne-Zeilinger state by stochastic local operations and classical communication, and discuss generalizations of our results to an arbitrary number of qubits.