To get a list of all of the Quick Queries in the system do an HttpGet to:
http://servername/ctDynamicsSLWebApi/api/quickQuery/

To call a Quick Query in the system with no filters do an HttpGet or HttpPost to:
http://servername/ctDynamicsSLWebApi/api/quickQuery/{queryViewName}
This will return our screen object. The object will contain the quick query's entry in the quick query table (myQueryView) and the results from the query in a dataset (myQueryResults)

To apply filters to the Quick Query do an HttpPost to:
http://servername/ctDynamicsSLWebApi/api/quickQuery/{queryViewName}
The function accepts a payload containing a filter variable containing an array of queryFilter objects. The queryFilter.name can be any valid column returned from the quick query. The queryFilter.comparisonType can be: =, <, >, <>, LIKE or any other valid sql comparison operator. eg:

{
    "filters":
    [
    {
        "name": "Customer ID",
        "value": "00149",
        "comparisonType": "="
    }    
    ,{
        "name": "Ship To ID",
        "value": "A000001003",
        "comparisonType": "="
    }    
    ]
}

The ctDynamicsSL SOAP interface includes the same functionality.
All methods for the SOAP interface can be found here.

Note: Both the REST and SOAP interfaces require the same standard headers for authentication as all other CTAPI web services.