Document:

<PAGE>

                                                                    EXHIBIT 10.6

                 SECOND AMENDMENT TO THE TAX SHARING AGREEMENT

                                 by and among

                                  AT&T CORP.,

                          LIBERTY MEDIA CORPORATION,

               for itself and each member of the Liberty Group,

                          TELE-COMMUNICATIONS, INC.,

                          LIBERTY VENTURES GROUP LLC,

                           LIBERTY MEDIA GROUP LLC,

                               TCI STARZ, INC.,

                             TCI CT HOLDINGS, INC.

                                      and

           each Covered Entity listed on the signature pages hereof,

                        dated as of September 24, 1999
<PAGE>

                 SECOND AMENDMENT TO THE TAX SHARING AGREEMENT
                 ---------------------------------------------

          This Second Amendment, dated as of September 24, 1999 (this "Second
Amendment"), to the Tax Sharing Agreement dated as of March 9, 1999, as amended
by the First Amendment (the "First Amendment") to the Agreement dated as of May
28, 1999 (the "Agreement") is entered into by and among AT&T Corp., a New York
corporation ("AT&T"), Liberty Media Corporation, a Delaware corporation
("Liberty"), for itself and on behalf of each member of the Liberty Group, Tele-
Communications, Inc., a Delaware corporation, Liberty Ventures Group LLC, a
Delaware limited liability company, Liberty Media Group LLC, a Delaware limited
liability company, TCI Starz, Inc., a Colorado corporation, TCI CT Holdings,
Inc., a Delaware corporation, each Covered Entity listed on the signature pages
hereof, and any entities which become parties to the Agreement pursuant to
Section 23 thereto.  Unless otherwise stated herein, capitalized terms used in
this Second Amendment shall have the meaning ascribed to such terms in the
Agreement.

          WHEREAS, the parties have entered into the Agreement, which governs
the sharing, allocation and reimbursement of federal, state, local and foreign
taxes by the members of the Common Stock Group and the Liberty Group;

          WHEREAS, the Capital Stock Committee of the AT&T Board of Directors
has adopted the resolutions attached as Exhibit A to the Second Supplement to
the Inter-Group Agreement, dated as of September 24, 1999 (the "Second
Supplement"), approving and authorizing the repurchase by AT&T from time to time
of up to 135 million shares of Liberty Media Group Tracking Stock (as such
number shall be adjusted from time to time to reflect stock splits, stock
dividends, stock combinations and similar events affecting the Liberty Media
Group Tracking Stock), in accordance with the terms and conditions set forth in
the resolution of the Capital Stock Committee and the Second Supplement (the
"Stock Repurchase Program");

          WHEREAS, the parties intend that certain Tax Items arising from or
related to the Stock Repurchase Program shall be considered Tax Items
attributable to the Liberty Group; and

          WHEREAS the parties now wish to amend the Agreement in certain
respects to clarify the intent of the parties with respect to sharing,
allocation and reimbursement of federal, state, local and foreign taxes by the
members of the Common Stock Group and the Liberty Group and to make such other
amendments, as provided herein;

          NOW, THEREFORE, in consideration of the mutual covenants contained
herein, the parties hereby amend the Tax Sharing Agreement and agree as follows:

          1.    The Agreement, as amended by the First Amendment, is amended by
inserting in Section 1: "`Repurchase Transactions' shall have the meaning set
forth in the Second Supplement."
<PAGE>

          2.    The Agreement, as amended by the First Amendment, is amended by
inserting in Section 3(d)(i) the words "the adoption by the Capital Stock
Committee of the AT&T Board of Directors of the resolutions attached as Exhibit
A to the Second Supplement or any action taken by AT&T or any other member of
the Common Stock Group in good faith in accordance with the terms of the Second
Supplement or Second Amendment in connection with the Stock Repurchase Program
or any Repurchase Transaction or" after the words "(except in the case of
clauses (i), (ii) and (iii), to the extent arising out of or relating to" and
before the words "actions taken by AT&T at the request of Liberty...."

          3.    The Agreement, as amended by the First Amendment, is amended by
inserting as Section 3(d)(xi):

          "(xi) Liberty Stock Repurchase Program.  Except as set forth in the
                --------------------------------
second sentence of this Section 3(d)(xi) with respect to the Excluded Buyback
Tax Items (as defined below) and in addition to (but without duplication of) any
Tax Item for which Liberty is responsible (other than pursuant to this Section
3(d)(xi)) under this Agreement, any Tax Item (including, without limitation, any
gain recognized under Code Section 311(b) and any fee or interest income)
arising from or relating to:  (I) any agreement required to be entered into by
AT&T in connection with any Repurchase Transaction or the negotiation, review,
execution or delivery of the Second Amendment or the Second Supplement; (II) any
Repurchase Transaction entered into in accordance with the terms of the Second
Supplement and the applicable Repurchase Notice (as defined in the Second
Supplement); (III) any action taken in good faith by AT&T or any member of the
Common Stock Group in accordance with the terms of the Second Supplement or the
Second Amendment; (IV) AT&T's performance of its obligations under the Second
Supplement (or the Second Amendment) relating to the Stock Repurchase Program in
accordance with the terms of the Second Supplement (or the Second Amendment);
(V) any transaction undertaken by any member of the Liberty Group in connection
with the Stock Repurchase Program, any Repurchase Transaction or any transaction
related thereto; (VI) any excess loss account in the stock of Liberty or any
other member of the Liberty Group; and (VII) any excess loss account in the
stock of any member of the Common Stock Group to the extent that such excess
loss account would not have existed but for the creation or increase in the
excess loss accounts, if any, in the stock of Liberty and the Covered Entities
(to the extent such stock in the Covered Entities is held directly or indirectly
by Liberty Ventures Group LLC or TCI), determined on a collective basis, shall
be for the account of the Liberty Group, and Liberty shall pay AT&T any Tax (or
any reduction in any Tax refund, credit or other benefit) attributable thereto.
Any Tax Item arising from or relating to: (A) any excess loss account in the
stock of any member of the Common Stock Group (except as provided in clause
(VII) above of this Section 3(d)(xi)) or (B) the disposition by any member of
the Common Stock Group in a Repurchase Transaction of Liberty Media Group
Tracking Stock held for the account of the Common Stock Group (the foregoing Tax
Items specified in this sentence shall be referred to as "Excluded Buyback Tax
Items"), shall be for the account of AT&T.

                                      -2-
<PAGE>

          4.   The words "the date hereof" shall be deleted in each place that
they appear in the Agreement and in lieu thereof shall be inserted the words
"March 9, 1999."

          5.   Except as otherwise expressly provided herein, the Agreement
shall continue in full force and effect without modification.

                                      -3-
<PAGE>

          IN WITNESS WHEREOF, each of the parties has caused this Second
Amendment to be executed by its respective duly authorized officer as of the
date first set forth above.

                             AT&T CORP.

                             By: /s/ Marilyn J. Wasser
                                 -------------------------------------
                             Name: Marilyn J. Wasser
                             Title:

                             LIBERTY MEDIA CORPORATION, for itself and for
                             each member of the Liberty Group

                             By: /s/ Gary S. Howard
                                 -------------------------------------
                             Name: Gary S. Howard
                             Title: Executive Vice President

                                      -4-
<PAGE>

Each of the Covered Entities listed below on this page hereby executes this
Second Amendment as a member of the Liberty Group to acknowledge that such
Person is bound by this Second Amendment as a member of the Liberty Group:

                              LIBERTY SP, INC.

                              By: /s/ Gary S. Howard
                                  ------------------------------------
                              Name: Gary S. Howard
                              Title: Executive Vice President

                              LIBERTY AGI, INC.

                              By: /s/ Gary S. Howard
                                  ------------------------------------
                              Name: Gary S. Howard
                              Title: Executive Vice President

                              LMC INTERACTIVE, INC.

                              By: /s/ Gary S. Howard
                                  ------------------------------------
                              Name: Gary S. Howard
                              Title: Executive Vice President

                                      -5-
<PAGE>

                               TELE-COMMUNICATIONS, INC.

                               By: /s/ Stephen M. Brett
                                   -----------------------------------
                               Name: Stephen M. Brett
                               Title:

                               LIBERTY VENTURES GROUP LLC

                               By: /s/ Stephen M. Brett
                                   -----------------------------------
                               Name: Stephen M. Brett
                               Title:

                               LIBERTY MEDIA GROUP LLC

                               By: /s/ Gary S. Howard
                                   -----------------------------------
                               Name: Gary S. Howard
                               Title: Executive Vice President

                               TCI STARZ, INC.

                               By: /s/ Stephen M. Brett
                                   -----------------------------------
                               Name: Stephen M. Brett
                               Title:

                               TCI CT HOLDINGS, INC.

                               By: /s/ Gary S. Howard
                                   -----------------------------------
                               Name: Gary S. Howard
                               Title: Vice President

                                      -6-<PAGE>

                                                                    EXHIBIT 10.7

                               THIRD AMENDMENT TO

                             TAX SHARING AGREEMENT

                                  by and among

                                  AT&T CORP.,

                           LIBERTY MEDIA CORPORATION,
                for itself and each member of the Liberty Group,

                           TELE-COMMUNICATIONS, INC.,

                          LIBERTY VENTURES GROUP LLC,

                            LIBERTY MEDIA GROUP LLC,

                                TCI STARZ, INC.,

                             TCI CT HOLDINGS, INC.,

                                      and

           each Covered Entity listed on the signature pages hereof,

                          dated as of October 20, 1999
<PAGE>

          This Third Amendment, dated as of October 20, 1999 (this "Third
Amendment"), to the Tax Sharing Agreement dated as of March 9, 1999, as amended
by the First Amendment (the "First Amendment") to the Tax Sharing Agreement
dated as of May 28, 1999 and the Second Amendment (the "Second Amendment") to
the Tax Sharing Agreement dated as of September 24, 1999 (the "Agreement"), is
entered into by and among AT&T Corp., a New York corporation ("AT&T"), Liberty
Media Corporation, a Delaware corporation ("Liberty"), for itself and on behalf
of each member of the Liberty Group, Tele-Communications, Inc., a Delaware
corporation, Liberty Ventures Group LLC, a Delaware limited liability company,
Liberty Media Group LLC, a Delaware limited liability company, TCI Starz, Inc.,
a Colorado corporation, TCI CT Holdings, Inc., a Delaware corporation, each
Covered Entity listed on the signature pages hereof, and each entity which
becomes a party to the Agreement pursuant to Section 23 thereto. Unless
otherwise stated herein, capitalized terms used in this Third Amendment shall
have the meaning ascribed to such terms in the Agreement.

          WHEREAS, the parties have entered into the Agreement which governs the
sharing, allocation and reimbursement of federal, state, local and foreign taxes
by the members of the Common Stock Group and the Liberty Group;

          WHEREAS, AT&T intends to acquire Ascent Entertainment Group, Inc., a
Delaware corporation ("Ascent"), in a transaction qualifying as a tax-free
reorganization under Section 368(a) of the Code pursuant to an Agreement and
Plan of Merger dated as of October 20, 1999 (the "Ascent Merger Agreement") for
and on behalf of the Liberty Group;

          WHEREAS, the parties intend that any Tax Items arising from or
relating to the Ascent Merger (as defined below), including any Tax Items of
Ascent or any of its direct or indirect assets or subsidiaries, shall be
considered Tax Items attributable to the Liberty Group except to the extent set
forth herein; and

          WHEREAS, the parties now wish to amend the Agreement in certain
respects to clarify the intent of the parties with respect to the sharing,
allocation and reimbursement of federal, state, local and foreign taxes by the
members of the Common Stock Group and the Liberty Group and to make such other
amendments, as provided herein;

          NOW, THEREFORE, in consideration of the mutual covenants contained
herein, the parties hereby amend the Tax Sharing Agreement as follows:

          1.        The Agreement is amended by inserting as Section 3(d)(xii):

          "(xii)    Ascent Merger.  Any Tax Item arising from or relating to
                    -------------
Ascent, Ranger Acquisition Corp. ("Ascent Merger Sub"), On Command Corporation,
COMSAT Corporation, or any of their respective direct or indirect subsidiaries
or affiliates (or any predecessor or successor of any of the foregoing under
applicable corporate, limited liability company, partnership or other
organizational law) (the "Ascent Entities"); the status of any member of the
Common Stock Group as the successor under Code Section 381 (or comparable
provision of state, local or
<PAGE>

foreign Tax law) to any of the Ascent Entities; any direct or indirect asset,
liability, business, investment or operation of any of the Ascent Entities; the
merger of Ascent Merger Sub with and into Ascent (the "Ascent Merger"), the
Ascent Merger Agreement, the Post-Merger Restructuring Transactions (as defined
in the Ascent Merger Agreement), the issuance of New Liberty Media Group
Tracking Stock in the Ascent Merger or any other transaction contemplated by the
Ascent Merger Agreement, the Third Supplement to Inter-Group Agreement dated
October 20, 1999 (the "Third Supplement"), or this Third Amendment, or any other
document to which the Company, Liberty or any of their respective Subsidiaries
(as defined in the Ascent Merger Agreement) or Affiliates (as defined in the
Ascent Merger Agreement) is a party that is referred to in the Ascent Merger
Agreement, the Third Supplement, and this Third Amendment or executed in
connection therewith (any of the foregoing Tax Items specified in this sentence
shall be referred to hereinafter as an "Ascent Tax Item"), shall be for the
account of the Liberty Group (except to the extent otherwise provided in this
Section 3(d)(xii) with respect to any Ascent Tax Item), and Liberty shall pay
AT&T any Tax (or any reduction in any Tax refund, credit or other benefit)
attributable thereto. Notwithstanding anything in the preceding sentence to the
contrary, any Ascent Tax Item shall be for the account of AT&T hereunder if, and
to the extent that, such Ascent Tax Item arises directly from and would not have
arisen but for (i) any inaccuracy in any of the representations by AT&T or
Ascent Merger Sub in the Officer's Certificate dated as of the closing date of
the Ascent Merger delivered by AT&T and Ascent Merger Sub in connection with the
opinions to be delivered pursuant to Sections 8.2(h) and 8.3(e) of the Ascent
Merger Agreement, (ii) any breach by AT&T or Ascent Merger Sub of any of their
representations or covenants in Sections 3.9, 3.10, 3.11, 5.3, 7.7, 7.12, 7.13
of the Ascent Merger Agreement or (iii) any breach by AT&T of any representation
or covenant in the Inter-Group Agreement (except in the case of clauses (i),
(ii) and (iii), to the extent arising out of or relating to the adoption by the
Capital Stock Committee of the AT&T Board of Directors of the resolutions
attached as Exhibit A to the Second Supplement to the Inter-Group Agreement,
dated as of September 24, 1999 (the "Second Supplement") or any action taken by
AT&T or any other member of the Common Stock Group in good faith in accordance
with the terms of the Second Supplement or Second Amendment in connection with
the Stock Repurchase Program (as defined in the Second Supplement) or any
Repurchase Transaction or actions taken by AT&T at the request of Liberty as
contemplated by Section 1.2(d) of the Third Supplement or otherwise in writing),
and AT&T shall pay to the applicable Governmental Authority or to Liberty any
Tax, and shall pay to Liberty any reduction in any Tax refund, credit or other
benefit that is for the account of Liberty hereunder, attributable thereto.

          2.   The Agreement is amended by (i) deleting in Section 9(b) the word
"or" and inserting in lieu thereof "," after the words "any Pre-Closing Taxable
Period" and before the words "(II) AGI for any taxable period . . ." and (ii)
inserting in Section 9(b) the words ", or (III) Ascent for any taxable period
ending on or prior to the date of the closing of the Ascent Merger or any
Subsidiary of Ascent during any such period for such period" after the words "or
any Subsidiary of AGI during any such period for such period" and before the
words "; provided, however, that (i) AT&T shall be entitled to participate . . .
 ."
<PAGE>

          3.   Except as otherwise expressly provided herein, the Agreement
shall continue in full force and effect without modification.
<PAGE>

          IN WITNESS WHEREOF, each of the parties has caused this Third
Amendment to be executed by its respective duly authorized officer as of the
date first set forth above.

                              AT&T CORP.

                              By:   /s/ Daniel E. Somers
                                    --------------------
                                    Name:  Daniel E. Somers
                                    Title: Senior Executive Vice President and
                                           Chief Financial Officer

                              LIBERTY MEDIA CORPORATION, for
                              itself and for each member of the Liberty Group

                              By:   /s/ Gary S. Howard
                                    ------------------
                                    Name:  Gary S. Howard
                                    Title: Executive Vice President and Chief
                                           Operating Officer
<PAGE>

Each of the Covered Entities listed below on this page hereby executes this
Third Amendment as a member of the Liberty Group to acknowledge that such Person
is bound by this Third Amendment as a member of the Liberty Group:

                              LIBERTY SP, INC.

                              By:   /s/ Gary S. Howard
                                    ------------------
                                    Name:  Gary S. Howard
                                    Title: Executive Vice President and Chief
                                           Operating Officer

                              LIBERTY AGI, INC.

                              By:   /s/ Gary S. Howard
                                    ------------------
                                    Name:  Gary S. Howard
                                    Title: Executive Vice President and Chief
                                           Operating Officer

                              LMC INTERACTIVE, INC.

                              By:   /s/ Gary S. Howard
                                    ------------------
                                    Name:  Gary S. Howard
                                    Title: Executive Vice President and Chief
                                           Operating Officer
<PAGE>

                              TELE-COMMUNICATIONS, INC.

                              By:   /s/ Daniel E. Somers
                                    --------------------
                                    Name:  Daniel E. Somers
                                    Title: Acting President

                              LIBERTY VENTURES GROUP LLC

                              By:   /s/ Stephen M. Brett
                                    --------------------
                                    Name:  Stephen M. Brett
                                    Title: Vice President and Secretary

                              LIBERTY MEDIA GROUP LLC

                              By:   /s/ Gary S. Howard
                                    ------------------
                                    Name:  Gary S. Howard
                                    Title: Vice President

                              TCI STARZ, INC.

                              By:   /s/ Stephen M. Brett
                                    --------------------
                                    Name:  Stephen M. Brett
                                    Title: Vice President and Secretary

                              TCI CT HOLDINGS, INC.

                              By:   /s/ Gary S. Howard
                                    ------------------
                                    Name:  Gary S. Howard
                                    Title: Executive Vice President and Chief
                                           Operating Officer

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