We formulate a covariant description of a relativistic qubit identified with an irreducible set of quantum spin states of a Majorana particle with a sharp momentum. We treat the particle’s four-momentum as an external parameter. We show that it is possible to define an interesting time evolution of the spin density matrix of such a qubit. This evolution is manifestly Lorentz covariant in the bispinor representation and unitary in the spin representation. Moreover, during this evolution the Majorana particle undergoes an uniformly accelerated motion. We classify possible types of such motions, and finally we illustrate the behaviour of the polarization vector of the Majorana qubit during the evolution in some special cases.