How to detect sample size for multiple groups in an AB test? I'm sure you know this sample size calculator by Evan Miller. Given necessary parameters, it outputs the efficient sizes for Control & Test groups.
But what if I have multiple Test groups? What should be their sample sizes?
Ideally, the solution should take multiple testing corrections into account, but it is not required. 
I would love to see derivations!
 A: The sample size calculations are the same as the standard RCT case when using multiple groups. Therefore, if we require ~600 users per group to detect a particular change in proportion with a given statistical power and significance level, if we have 5 variation we will require ~3,000 users. What changes is exactly the point you raised about multiple correction.
Assuming we do not have a multitude of independent A/B test (e.g. <10) Bonferroni correction is actually pretty decent. For example, the probability of at least one significant result when we have 5 tests and we use Bonferroni correction to control for multiple comparison is: $1-P(\text{No significant result})$ $\Rightarrow $ $1 - (1 - \frac{0.05}{5})^5 $ $=$ $0.0490$; for practical terms this (new) family-wise error rate based on the adjusted level of $\frac{\alpha}{k}$ is usually fine. The power will suffer though because the associated $z_{\alpha/2}$ will immediately get much more extreme. It would be uncommon to go from 0.80 to 0.50 just by correcting for 5 tests. Chow et al. (2008) "Sample Size Calculations in Clinical Research" Ch. 4: Large Sample Tests for Proportions is a really good resource on the matter if you wish to explore this behaviour further, as well as the package pwr in R.
When we have more comparisons the Bonferroni correction is very likely too conservative (i.e. our Type II error rate gets too high). This leads to requiring a potentially unreasonably large sample size to detect a fixed difference with given statistical power. The main option is to usually use FDR methods - control for the False Discovery Rate (which is a huge subject). As an immediate middle ground we might want to use Holm-Bonferroni method, which is just as general as Bonferroni but less conservative. Again, we can explore this behaviour with R using the stats::p.adjust which is immediately available in every R installation.
Finally, notice that in the context of A/B test for a website we often look into multi-armed bandit (MAB) optimization. MAB is not focusing on statistical significance but rather on exploitation of traffic. It relates much closer to Thompson Sampling rather than an RCT; Russo et al. (2017) "A Tutorial on Thompson Sampling" is probably the most complete
tutorial on matter.
