Current Case Execution API endpoint can be queried to retrieve the Case Execution (GET URL/case-execution). The answer consists the following JSON structure. For example:
[{id: "3a93f7ee-5057-11e4-951d-58ff20524153",
caseInstanceId: "3a930d8d-5057-11e4-951d-58ff20524153",
caseDefinitionId: "Werkstatt_besuchen:1:31cf516b-5055-11e4-951d-58ff20524153",
activityId: "PIhumanCreateCustomerOrder",
activityName: "Kundenauftrag erfassen",
parentId: "3a930d8d-5057-11e4-951d-58ff20524153",
enabled: true,
active: false,
disabled: false
},

{...}]

This is perfectly fine, but if you are building a case UI, it is very helpful to know the type of the execution (HUMAN TASK, CASE TASK, PROCESS TASK, STAGE, CASE) and the description. Since this information is available, it would be very nice to access to it via API:

[{id: "3a930d8d-5057-11e4-951d-58ff20524153", caseInstanceId: "3a930d8d-5057-11e4-951d-58ff20524153", caseDefinitionId: "Werkstatt_besuchen:1:31cf516b-5055-11e4-951d-58ff20524153", activityId: "caseDefCarShop", activityName: null, parentId: null, enabled: false, active: true, disabled: false, type: "HUMANTASK", description: "Erfassen von Kunden- und Fahrzeugdaten sowie des Kundenwunsches"}, {...}

]