In fiscal year 2025, I investigated the braid group action on the Stokes matrices associated with the tt*-Toda equations. This topic has been studied by Cotti, Dubrovin, Guzzetti, and others in connection with problems in quantum cohomology. In that context, it is closely related to the analysis of quantum differential equations.From an algebraic viewpoint, Hertling and Larabi have carried out a detailed study of this problem. Their work is also motivated by singularity theory and considers not only Stokes matrices but also braid group actions on objects known as distinguished bases. Roughly speaking, a distinguished basis can be regarded as a "triangular basis" of the solution space of a quantum differential equation.In this research, we defined distinguished bases for the “quantum differential equations” arising from the tt*-Toda equations and investigated their properties. The results on the tt*-Toda equations used in this work are based on a series of results by Guest, Its, and Lin.First, it extends the previous work of Cotti–Dubrovin–Guzzetti by incorporating the perspective of distinguished bases. In particular, we described the action of a certain element of the braid group in terms of quantities known as holomorphic data, which characterize global solutions of the tt*-Toda equations. Using this description, we computed the periods of these braid group actions explicitly in terms of the holomorphic data.Second, this work provides an analytic approach to problems that were previously studied by Hertling and Larabi from a primarily algebraic perspective. While their work focused on the case where the Stokes matrices are integer matrices, our research studied more general cases where the matrices are not necessarily integral, and examines the braid group action in this broader setting.The results of this research were presented at a workshop held in Takamatsu in November 2025, and at another workshop held at the University of Mannheim, Germany, in January 2026.