For more than a century, quantum mechanics, the theory of the infinitely small, keeps offering surprises through its notoriously counter-intuitive predictions. As an example, any measurement performed on a quantum system intrinsically disturbs its state, so that there can exist incompatible measurements that cannot be performed at the same time. A famous instance of this property is quantified in Heisenberg's uncertainty principle, which limits the simultaneous accuracy on the position and the momentum. In this manuscript, I examine quantitatively this property in the case of quantum systems with a finite but potentially large number of degrees of freedom. Beyond quantum bits, the results to this regard were quite rare, due to the difficulty to visualise high-dimensional states, whose number of parameters grows quite fast. However, by developing new universal computational methods or by exploiting abstract structures likely to be generalised, I could start to clear the way towards the understanding of the complexity arising in high dimensions. Given the natural link between incompatibility and quantum steering, the results were quite useful in this field and I finish by discussing these applications. This manuscript articulates several scientific articles published during the course of my thesis. In an attempt to smoothly present the ones sharing a common problematic, I handpicked the results to be covered and often reduced the level of detail. Complete proofs, additional results, and other publications can be found in the original works, given in appendix.