  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
phenomenons: 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.
