In the last few years the non-commutative moment inequalities have received a considerable attention in matrix analysis and operator theory. This phenomenon originates on the one hand in studies on the extreme properties of the standard deviation in quantum information theory, and on the other hand, in the recent concept and developments of quantum metric spaces. This research project will investigate trace inequalities in matrix algebras. Particular attention will be paid on moment inequalities for matrices with a special emphasis on their counterparts in operator algebras. It will focus on determining the best upper and lower bounds for higher order central moments in matrix algebras. This will be followed by a study of the relatively new concept of Leibniz seminorms in Banach algebras. The project will pursue a research on the strong Leibniz property of central moments in non-commutative probability spaces as well as in the classical ones in order to solve the recent question whether every centered moment has the strong Leibniz property or not.