Secondary halo bias, commonly known as assembly bias, is the dependence of halo clustering on a halo property other than mass. This prediction of the Lambda-Cold Dark Matter cosmology is essential to modelling the galaxy distribution to high precision and interpreting clustering measurements. As the name suggests, different manifestations of secondary halo bias have been thought to originate from halo assembly histories. We show conclusively that this is incorrect for cluster-size haloes. We present an up-to-date summary of secondary halo biases of high-mass haloes due to various halo properties including concentration, spin, several proxies of assembly history, and subhalo properties. While concentration, spin, and the abundance and radial distribution of subhaloes exhibit significant secondary biases, properties that directly quantify halo assembly history do not. In fact, the entire assembly histories of haloes in pairs are nearly identical to those of isolated haloes. In general, a global correlation between two halo properties does not predict whether or not these two properties exhibit similar secondary biases. For example, assembly history and concentration (or subhalo abundance) are correlated for both paired and isolated haloes, but follow slightly different conditional distributions in these two cases. This results in a secondary halo bias due to concentration (or subhalo abundance), despite the lack of assembly bias in the strict sense for cluster-size haloes. Due to this complexity, caution must be exercised in using any one halo property as a proxy to study the secondary bias due to another property.
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