Correlation intractable function ensembles were introduced in an attempt to capture the “unpredictability” property of a random oracle: It is assumed that if R is a random oracle then it is infeasible to find an input x such that the input-output pair (x;R(x)) has some desired property. Since this property is often useful to design many cryptographic applications in the random oracle model, it is desirable that a plausible construction of correlation intractable function ensembles will be provided. However, no plausibility result has been proposed. In this paper, we show that proving the implication, “if one-way functions exist then correlation intractable function ensembles exist”, is as hard as proving that “3-round auxiliary-input zero-knowledge Arthur-Merlin proofs exist only for trivial languages such as BPP languages.” As far as we know, proving the latter claim is a fundamental open problem in the theory of zero-knowledge proofs. Therefore, our result can be viewed as strong evidence that the construction based solely on one-way functions will be impossible, i.e., that any plausibility result will require stronger cryptographic primitives.