We will present some recent results. First, we show that some sets of quantum observables are unique up to an isometry and have a contextuality witness that attains the same value for any initial state. These two properties enable to certify them using the statistics of experiments with sequential measurements on any initial state of full rank. Second, we show that the following statements are equivalent: (i) A quantum correlation p is in a face of the nonsignaling polytope that does not contain local points. (ii) p has local fraction zero, i.e., p has full nonlocality (FN). (iii) p provides an all-versus-nothing (AVN) or Greenberger-Horne-Zeilinger-like proof of nonlocality. (iv) p is a pseudo telepathy (PT) strategy. These equivalences imply that a long-standing question of what is the simplest example of bipartite FN/AVN/PT has fundamental relevance. We will answer this question.