Since the beginning of quantum mechanics, many puzzling phenomena which distinguish the quantum from the classical world, have appeared such as complementarity, entanglement or contextuality. All of these phenomena are based on the existence of non-commuting observables in quantum mechanics. Furthermore, theses effects generate advantages which allow quantum technologies to surpass classical technologies. In this lecture note, we investigate two prominent examples of these phenomena: complementarity and entanglement. We discuss some of their basic properties and introduce general methods for their experimental investigation. In this way, we find many connections between the investigation of complementarity and entanglement. One of these connections is given by the Cauchy-Schwarz inequality which helps to formulate quantitative measurement procedures to observe complementarity as well as entanglement.