No, it is not possible to use any of the Quantities renderers (Graduated color, Graduated symbols, Proportional symbols) to symbolize dates in ArcMap. This is a known limitation when dealing with a date field.
As an alternative to symbolizing a date field, the unique values renderer can be used to symbolize different attribute values, such as timeframes and dates. The following procedure illustrates how to symbolize a date field using unique values by grouping the dates.
1. In ArcMap, right-click the desired layer in the Table Of Contents and select Properties.
2. In the Layer Properties dialog box, select the Symbology tab. In the left pane under 'Show', click Categories and select 'Unique values'.
3. Under the Value Field, select the date field from the drop-down arrow, and select Add All Values to list the dates.
Add All Values allows the selection of subset of field values to be included as categories in the layer display.
The dates are displayed based on the Windows setting. Change the date format to ensure the date is listed based on the desired sequence (for example, 'YY-MM-DD').
4. To group the dates (for example, based on year), press and hold down the Ctrl key on the keyboard and click the dates. Right-click any of the highlighted dates and select Group Values.
Grouping the dates ensure each group is assigned with the same color.
If a quantity is needed for data analysis, such as the total amount of records within a certain group in the applicable date field, use the 'Count' values on the Symbology tab in Layer Properties.
5. If more groups must be created, repeat step 4. Continue repeating step 4 until the desired number of groups is achieved.
6. Click Apply and OK. The date field is now symbolized using Unique Values.