Having recently struggled with a key date prompt in a BO universe built on a BEx query ... I'm posting here some samples and screenshots of the outcome as I didn't find/couldn't understand documentation on this .... 
The universe wizard in my case didn't generate something usable out of the box, so I had to tweak it a bit 
First one, how to show it as a datepicker (sounds strange ...) 
 
 The corresponding code for the prompt is: 
<filter key="[YV_DAY]"><condition operatorcondition="Equal"><constant tech_name=" @Prompt('Data Cardine (formato YYYYMMDD)','D',,mono,free)"></condition></filter> 
Then if you want it to be a simple, no frills, single value prompt 
 <filter key="[YV_DAY]"><condition operatorcondition="Equal"><constant tech_name="@Prompt('Data Cardine (formato YYYYMMDD)','A',,mono,free)"></condition></filter>
 
<filter key="[YV_DAY]"><condition operatorcondition="Equal"><constant tech_name="@Prompt('Data Cardine (formato YYYYMMDD)','A',,mono,free)"></condition></filter> 
And if you want the prompt to “forget” the previous value when the query is refreshed: 
<filter key="[YV_DAY]"><condition operatorcondition="Equal"><constant tech_name="@Prompt('Data Cardine (formato YYYYMMDD)','A','Data\LovData CardineBase',mono,free, not_persistent)"></condition></filter> 
Also see note 
1407631 and keep up with the updates on BW and BO side, IMHO it’s not a mature solution yet. 
Note 
1142664 - MDX: Composite SAP note about performance improvements 
Note 
838800 - Composite SAP Note for consulting notes about MDX/OLAP BAPIs 
Note 
1156101 - MDX: Composite SAP note for incorrect data 
Another useful one
Note 
1567394 - MDX: Consulting note for sort sequence