Solution can be installed on many environments - QA, Staging, Prod. In every of these environments Sitecore instances have different roles - CD, CM, Job, Publishing and etc.
Should I install Powershell on all these instances if I use SXA?
SPE is only required for environments that use the Sitecore interface. If you are going to use the Content Editor or Experience Editor then install the module.
Essentially only the CM needs SPE.
Currently Sitecore PowerShell Extensions is used in SXA for the following things:
Although it is requirement your instance will still work fine if you are not going to use script automation.
Normally you should be able to do everything without PowerShell however site scaffolding is too complicated to do it manually.
Bear in mind that in the future there might appear more integrations with SPE do you should probably monitor those to eliminate possible problems on your instances without module.
Yes is mandatory to install Powershell if you want to have all facilities from SXA. On the installation guide you wil find: Before installing SXA, ensure that you have the following:
Sitecore Experience Platform 8.2 initial release or Sitecore Experience Platform 8.1 Update-3
Sitecore PowerShell extensions
o Sitecore XP 8.1 – full 4.1 or later
o Sitecore XP 8.2 – full 4.1 or later
For example if you create tenants or you want to export site behind is used Powershell.