One of the most promising features of topological matter is the presence of helical ballistic edge states, pure one-dimensional propagating electronic states protected from backscattering by spin-momentum locking. When coupled to superconducting electrodes, a supercurrent is carried by topological Andreev bound states (ABS) also presenting spin-momentum locking. However, we are still far from a complete understanding of the role of the spin degree of freedom and the parity conservation. A major issue is to find unambiguous experimental signatures of the topological protection. We propose to tackle this problem through measurements of the dynamics and relaxation mechanisms of such states, which is still poorly explored experimentally. Our general idea is to develop an ultrasensitive magnetic field sensor by combining cryogenic amplifiers adapted to giant magneto-resistive (GMR) sensors, both homemade. We plan to detect fluctuations of the supercurrent at equilibrium in topological material coupled to superconducting electrodes. These current fluctuations originate from thermal excitation of ABS on a time scale given by the inelastic relaxation time, which can be as large as few milliseconds. This makes possible the real time detection of supercurrent fluctuations. We thus propose to improve the bandwidth and sensitivity of GMR detector up to the MHz range in order to detect real time fluctuations of occupation of the topological ABS. We will use different topological materials (Bi nanowires, WTe2 and Bi4Br4) whose fabrication we master and on which we have evidenced edge states. Eventually, such a high sensitivity should allow us to detect persistent current created by 1D loops of helical edge states in topological systems without superconducting contacts. This would represent one of the most relevant and direct evidence of the topological states.