ally cultivated audiences, physicists have been playing with entanglement for about 20 years. By means of increasingly sophisticated tools making it possible to manipulate sin- gle atoms, electrons, or photons, physicists are now beginning, literally, to "engineer" entangled states of matter. The Holy Grail they are after is building a practical toolbox for quantum entanglement. It is not clear at present which approach may show efficient, resilient, and environment-insensitive entanglement, while at the same time remaining "observable" and, furthermore, lending itself to external manipulation. To darken the picture, it is not at all clear either what could be the maximum size of an entangled system. Such questions come close to the actual definition of the boundary between the quantum and classical worlds, as emphasized by the famous Schrödinger's cat paradox. We may find ourselves in a situation similar to that of solid-state physics after World War II: quantum physics and many solid-state physics concepts were duly established, but the transistor remained yet to be invented. To the same extent that the revolution- ary concepts of electronic wave functions, band theory, and conductivity led to the development of modern electronics and computers, the concept of entanglement, which stands at the core of quantum information, is now waiting for a revolutionary outcome. The parallel evolution between the different constitutive elements of entangled sys- tems indeed offers huge opportunities to build radically new computing machines, with unprecedented characteristics and performance. What does entanglement have to do with complexity? Whereas basic mechanics laws can predict the trajectory of a ball, the oscillation period of a pendulum, the lift of a plane, complexity characterizes systems where the overall properties cannot be derived from that of the constituent subsystems. For the philosopher Edgar Morin, it is not the number of the components that defines the complexity of any system. More components certainly c