Correlation vs. causality Correlation does not imply causation.
Causation does imply correlation but not necessarily linear correlation.
...
So does correlation imply high-order causation?
If A and B are correlated, is it always possible to find K={K1 K2 ... Kn} variables such that 
A~K1, K1~K2, ...,Kn-1~Kn and Kn~B

where ~ denotes causality.
In other words, if there is no causality in a correlation, does a common-causal variable(s) always exist?
 A: You can simply have an omitted variable which has a causal impact on A and a causal impact on B. In that case, you will have a correlation between A and B but no "high-order causation". This issue is known in econometrics as the omitted variable bias. It is discussed in most econometrics textbooks. See for instance Cameron and Trivedi Microeconometrics. For a more advanced discussion see Judea Pearl's book on causality. 
A: No.
For example, we can find a correlation between global average temperature and the world population of pirates ("Arrrgh!"), but nobody would suggest there is any sort of causality involved:

Edit:
Okay, the pirates vs global temperatures example is not a very good one here. Because (a) the x-axis of the chart is highly distorted, and (b) we could probably actually stretch to find an omitted variable (like some measure of industrialization). 
A better example was given in answer to a previous question on CV:

A: If two events $A$ and $B$ are correlated given some observations $Z$ and interventions $Y$ in every context, then there must some causal mechanism mediating that correlation, which formally means that $A$ and $B$ are not d-separated by $Y$ and $Z$.
However, it's not clear what your notation means.   Examples of non-d-separation include

*

*$A$ causes $B$ (directly or indirectly),

*they may have a common cause, and

*they may both be causes of $Z$ or its ancestors.

