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.