  Quantum correlations are at the heart of many applications in quantum
information science and, at the same time, they form the basis for discussions
about genuine quantum effects and their difference to classical physics. On one
hand, entanglement theory provides the tools to quantify correlations in
information processing and many results have been obtained to discriminate
useful entanglement, which can be distilled to a pure form, from bound
entanglement, being of limited use in many applications. On the other hand, for
discriminating quantum phenomena from their classical counterparts,
Schr\"odinger and Bell introduced the notions of steering and local hidden
variable models. We provide a method to generate systematically bound entangled
quantum states which can still be used for steering and therefore to rule out
local hidden state models. This sheds light on the relations between the
various views on quantum correlations and disproves a widespread conjecture
known as the stronger Peres conjecture. For practical applications, it implies
that even the weakest form of entanglement can be certified in a semi-device
independent way.
