  Reliable generation and measurement of triphoton states has yet to be
achieved in laboratory. We give an overview of the problems in generating and
measuring triphoton quantum states and analyze several protocols of quantum
measurements, which allow for high precision of reconstruction when sizes of
available statistical data samples are limited. The tomography procedure under
investigation is based on root approach to state estimation. In particular, we
use the generalized Fisher information matrix to assess the accuracy of the
quantum state parameters measurement. We use tomographic protocols, based on
the symmetry of the Platonic solids. We demonstrate the capability to
reconstruct triphoton quantum states with precision close to the maximum
achievable value allowed by quantum mechanics
