A central goal of explainable artificial intelligence (XAI) is to improve the trust relationship in human-AI interaction. One assumption underlying research in transparent AI systems is that explanations help to better assess predictions of machine learning (ML) models, for instance by enabling humans to identify wrong predictions more efficiently. Recent empirical evidence however shows that explanations can have the opposite effect: When presenting explanations of ML predictions humans often tend to trust ML predictions even when these are wrong. Experimental evidence suggests that this effect can be attributed to how intuitive, or human, an AI or explanation appears. This effect challenges the very goal of XAI and implies that responsible usage of transparent AI methods has to consider the ability of humans to distinguish machine generated from human explanations. Here we propose a quantitative metric for XAI methods based on Turing's imitation game, a Turing Test for Transparency. A human interrogator is asked to judge whether an explanation was generated by a human or by an XAI method. Explanations of XAI methods that can not be detected by humans above chance performance in this binary classification task are passing the test. Detecting such explanations is a requirement for assessing and calibrating the trust relationship in human-AI interaction. We present experimental results on a crowd-sourced text classification task demonstrating that even for basic ML models and XAI approaches most participants were not able to differentiate human from machine generated explanations. We discuss ethical and practical implications of our results for applications of transparent ML.