  The outcomes of local measurements on entangled quantum systems can be
certified to be genuinely random through the violation of a Bell Inequality.
The randomness of the outcomes with respect to an adversary is quantified by
the guessing probability, conditioned upon the observation of a specific amount
of Bell violation or upon the observation of the entire input-output behavior.
It has been an open question whether standard device-independent randomness
generation protocols against classical or quantum adversaries can be
constructed on the basis of any arbitrary Bell inequality, i.e., does there
exist a Bell inequality for which the guessing probability is one for any
chosen input even upon observing the maximal violation of the inequality? A
strengthened version of the question asks whether there exists a quantum
behavior that exhibits maximum non-locality but zero certifiable randomness for
any arbitrary input. In this paper, we present an affirmative answer to both
questions by constructing families of $n$-player non-local games for $n \geq 2$
and families of non-local behaviors on the quantum boundary that do not allow
to certify any randomness against a classical adversary. Our results show the
existence of a form of bound randomness against classical adversaries,
highlighting that device-independent randomness and quantum non-locality can be
maximally inequivalent resources.
